TRANSACTIONS of theAMERICAN MATHEMATICAL SOCIETYVolume 344, Number 1, July 1994
GENERALISED CASTELNUOVO INEQUALITIES
LIAM A. DONOHOE
Abstract. Given a Riemann surface of genus p , denoted by Xp , admitting
j linear series of dimension r and degree n Accola derived a polynomial
function f(j, n, r) so that p < f{j, n,r) and exhibited plane models of
Riemann surfaces attaining equality in the inequality. In this paper we provide
a classification of all such Xp when r > 6 . In addition we classify curves,
Xp , of maximal genus when Xp admits two linear series which have a common
dimension but different degrees.
Introduction
In his paper [3] Castelnuovo produced a bound on the genus of an algebraic
curve in terms of n, the degree, and r, the dimension, of a complete, sim-
ple, fixed point free, linear series on the curve. He then exhibited the curves
which possessed maximal genus. In his paper [1] Accola generalised the idea
by allowing the curve to admit several linear series of the same degree and di-
mension and then set himself the task of answering the questions; is there an
upper bound on the genus of the curve in terms of the number of linear series
and their degree and dimension and (if it exists) is this bound ever attained?
In his paper he produced a bound and then exhibited plane models of curves of
maximal genus. We shall call curves of maximal genus generalised Castelnuovo
curves. Martens [9], Coppens [5, 6], and Coppens and Kato [7] have considered
the problem for the case r = 1. Coppens gives a classification of Castelnuovo
curves possessing several gj 's in [6]. In this paper we consider the problem for
the case r > 6.In this section we introduce the arguments used to produce the respective
upper bounds and establish some notation and fundamental ideas. All of these
are contained in [1]. In §1.1 and §1.2 we investigate the nature of the rela-tionship holding between several grn 's on a generalised Castelnuovo curve. In
1.2.1 we prove that if grn and hr„ are two such linear series then
K = g<n + g\-2!x-22
where g\ is a pencil imposing two conditions on grn and hrn and 2!¡ are fixed
divisors. From arguments used originally by Castelnuovo the model for Xp in
Received by the editors August 28, 1991 and, in revised form, March 11, 1993.
1991 Mathematics Subject Classification. Primary 14H99; Secondary 30A46.The author thanks the mathematics department of Brown University for its hospitality during
the preparation and revision of this paper. The author also wishes to acknowledge the many helpful
suggestions, comments, and corrections made by the referee to the original version of this paper.
©1994 American Mathematical Society0002-9947/94 $1.00+ $.25 per page
217
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218 L. A. DONOHOE
Pr lies on a rational normal surface scroll, the g\- corresponds to the ruling
on the scroll and the 31 i are the singularities of the model. In §2 and §3we perform the classification proper and we exhibit some plane models of the
curves in §4. Section 5 investigates the situation when Xp admits grn and hrn+p
and has maximal genus.
Definition 0.0. If grn and gsm are two complete linear series on a Riemann
surface Xp , with s <r and m < n we say that gsm imposes t linear conditions
on grn if there is a complete linear series grnZ'm so that grn = gsm + grn~Jm.
This means that if 3 is a general divisor of gsm containing m distinct points
then there are t distinct points of 3 , xx, ... , xt so that grn - (xx H-h xt)
contains the other points of 3 among its fixed points. Also xx, ... , xt imposeindependent conditions on grn, in the following sense: for each k there is a
divisor in grn containing all of the x, except for Xk -
We intoduce two results dealing with the general position of points on divisors
in linear series in projective space.The first one, which may be found in Coolidge
[4, Theorems 39, 43] is
Theorem 0.1. Let grn be a simple complete linear series without fixed points.
Then the general divisor 3 of grn is made up of n distinct points, and further-
more, any r points of 3 impose independent conditions on grn .
The following generalisation of the preceding result is due to Accola [1].
Recall that if 3 and % are two divisors, then their greatest common divisor
is denoted by (3, %).
Theorem 0.2. Suppose that grn and gsm are two linear series without fixed points
so that r > s. If both series are composite, suppose that there is no involution
of which both are compounded. Then, for the general divisor 3 in grn and any
divisor á? in gsm, the degree of (3, &) < s.
Using these general position theorems we now discuss a notion due to Castel-
nuovo. Let g„\, ... , grn\ be j fixed point free linear series on Xp . If grn\ = grn\
for some i > 2 then we assume that grn\ is simple, if all the grr/i are distinct
then we just assume that no two are composed of the same involution. Let
g§ - Yjgn¡ > and let g*\ = gfj + grn\ ■ We obtain a lower bound on Rx in the
following way. We notice firstly that Rx - R corresponds to the number of
conditions that g'n\ imposes upon g*1 . Let t¡ = r, - 1 if g„\ = grn\ and let
?, = min(rx, r¡) otherwise. Then, provided that nx > 1 + Yj¡=\ U • tne general
position theorem tells us that grn\ imposes at least 1 + ]££_. i, conditions on
gff1 and we have our lower bound on Rx.
Using Castelnuovo's method, Accola proves in his paper that if, rx > r2 >
■■ ■ > r, and the integers «, and /, satisfy the condition,
;'
ni > 53 krk - U + ! V/,k>i
then the linear series
rVo = hgr„\ + --- + ljgni
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GENERALISED CASTELNUOVO INEQUALITIES 219
has dimension at least
**,....r,././,) = ¿«>í|±ilr,-M|zil + I:E;,V,7=1 k i<k
The condition on the «, 's is neccessary because g„. cannot impose more than
n¡ conditions on W0 . An important feature of Accola's estimate of the lower
bound on the dimension of W0 in the case where rx = r2 = ■■ ■ — r¡■, is that it
is independent of the order in which the addition takes place.
We turn our attention to the situation where Xp admits j simple, complete,
special linear series of dimension r and degree n. We find a bound on the
genus of Xp by adding up multiples of the linear series until the resulting
linear series is nonspecial (always assuming that Accola's bound is attained,
and that Castelnuovo's method may be applied at each stage), then we can use
the dimension and degree of the nonspecial linear series to calculate p . Curves
whose genus attains this upper bound are the subject of this paper. Let us
point out that in the case where all the linear series have the same degree and
dimension, Accola's formula for the dimension is maximised when |/, —lj\< 1Vi, j ; for example, ?>grn + hr„ has dimension at least lOr - 3, while 2grn + 2hrn
has dimension at least 10r - 2.
1
This section will introduce several results necessary for our classification of
generalised Castelnuovo curves. Unless otherwise stated, we will assume our
grn 's to be complete, special, base point free, and simple. Wherever it appears,
the pencil g\- will be assumed to be base point free. Finally we shall assumethat r > 6.
1.1.
Remark 1.1.1. In this section we shall always assume that when we add multi-
ples of two or more linear series together that we can apply Accola's argument
to produce a lower bound on the dimension of the resulting linear series. Forexample, if we add two distinct grn 's and the resulting linear series has dimen-
sion 3r, then the condition n>3r will be understood. The reader is asked tobear this in mind in what follows.
We recall that in Castelnuovo's original argument for a Riemann surface
possessing one grn , it was critical for him that the dimension of 2grn equal
3r - 1. If r ^ 5 it was precisely this fact that allowed him to deduce thatthe image of the curve Xp under the map to Pr defined by grn lies on a
rational normal surface scroll FcPr [8]. If r = 5, then the curve may lie
on the Veronese surface in P5. Moreover, Castelnuovo was able to deduce the
existence of a g\ on the curve cut out by the rulings of the scroll. Finally, since
the hyperplanes in Pr cut out grn , he could also deduce that g\. imposes twoconditions on grn.
Lemma 1.1.2. If a Riemann surface Xp admits two complete linear series grn and
hrm, and r > 6 such that W — 2grn + hrm has dimension 6r - 1, the minimal
possible, then Xp possesses a gj- imposing two conditions on grn .
Proof. Since W has minimal dimension it follows that 2grn has minimal di-
mension and the result follows from Castelnuovo's observation. D
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220 L. A. DONOHOE
We quote, without proof, three results due to Accola [1] that we shall use
extensively throughout this paper.
Lemma 1.1.3. Suppose that grn is simple and gsm is any other linear series,
possibly composite, so that r > s and n > r + s — 1. Then the dimension of
grn + gfn /5 at least r + 2s.
Lemma 1.1.4. Suppose that grn and gsm are both simple, are without fixed points,
that n >m, r > s, and that n>4s-2. Suppose further that s > 2 and that the
dimension of grn + gsm is exactly r + 2s, then there are precisely three possibilities
for s and m :
(i) 5 = 2, r = 5, and g* = 2g2m .
(ii) s = r-2 and grn = gr~2 + g\_m .
(iii) s = r - 1 and grn = gr„x + g°n_m .
We follow Accola in remarking that if grm is simple and grm + gsm has di-
mension r + 2s, and r ^ 5 , 3 < s < r - 3 , then gsm is composite. The final
result we quote is the following .
Lemma 1.1.5. Suppose grn is simple and without fixed points, gsm is composite
and without fixed points, and that the dimension of grn + gsm is equal to r + 2s.
Then, there exists a complete gT, with sT = m so that gsm = sgj. Moreover,
g\- imposes precisely two conditions on grn .
The series of results quoted above are foundational to our analysis. In the
following two lemmas, we use them to describe completely the nature of grn
when 2grn has dimension 3r - 1 and grn - g\ is composite.
Proposition 1.1.6. Suppose that Xp admits a complete, simple linear series grn
and that 2grn has dimension 3r - 1. Let g\. be the linear series that imposes
two conditions on grn . Then grn + [gn - gf] has dimension 3r - 4.
Proof. Note that g} imposes at least three conditions upon 2grn and that by
assumption, this linear series has dimension 3r - 1. But 3r - 4 is the minimumpossible dimension for grn + [grn - gj], so we are done. D
Corollary 1.1.7. If grn is a complete simple linear series so that 2grn has dimen-
sion 3r - 1 and grn - g\- is composite (where g\- imposes two conditions on
grn), then there exists an effective divisor, 3 (possibly the zero divisor), so that
grn = (r-\)gxT+3.
Proof. By Proposition 1.1.6 and Lemma 1.1.5, we see that grnz}T is com-
pounded of a gg which imposes two conditions on grn , i.e.,
grn = (r-2)gxs + gxT + 3
where 3 is the fixed divisor of gr~2T. Our task is to show that gxs = g\.
Suppose that g\ and g\ are different, then g\ + g}¡ has dimension at least
three. Therefore, rewriting W = (r - 2)gxs + g\ as (r - 3)gj + (g\ + gxs) and
using the fact that g\ imposes at least two conditions on g\ + g^ we see that
W has dimension at least 3 + 2(r - 3) = 2r - 3 . Since r > 6, this number is
greater than r which is a contradiction. Therefore, gj-gg- U
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GENERALISED CASTELNUOVO INEQUALITIES 221
If Xp admits a grn so that 2grn has dimension 3r - 1, then its image in
Pr lies on a rational normal surface scroll Y. The model will be determined
in part by the degree of the gj which imposes two conditions upon the linear
system cut out on Xp by hyperplanes. We know of at least one gj on Xp,
the linear system cut out on image of Xp by the lines of the ruling on Y.
Conceivably, Xp could admit many pencils which impose two conditions on grn
and consequently, possess many distinct models. However, using an elementary
argument, we can show that there is only one such pencil.
Lemma 1.1.8. // Xp admits a complete, simple gr„ so that 2grn has dimension
3r - 1, then the gf imposing two conditions upon grn is unique.
Proof. Suppose that Xp admits two distinct pencils, g\ and gj, with both
imposing two conditions on grn . Our proof is split into two parts.
(a) Suppose that one of the series grn- g$, grn- g\- is simple, for argument's
sake let us assume that the latter series is simple. We have the relationships
grn = grnZ2s + gs and grn = grnz\ + gxT.
We add these expressions for grn and compare dimensions. By assumption, 2grn
has dimension 3r-l. By Lemma 1.1.3, grnZ2T-\-grnZ2s has dimension 3r-6+e.
Since gj. and gj are distinct, gj- + gj has dimension 3 +p. Applying 1.1.3
again, the right-hand side has dimension 3r + e + 2p + px, where e, p, and pxare all nonnegative. This is a contradiction.
(b) Suppose that both grn - g\- and gr„ - gj are composite then there exist
divisors & and S so that grn = (r - l)gj. +&> and g'n = (r - l)g\ +¿f. We
consider2grn = (gs + gxT) + (r- 2)gx + (r- 2)gxT + &+<?>.
The linear series g\- + gj has dimension at least three and both gj and gj
impose at least two conditions on it therefore, W = (gj + gf) + (r - 2)gj has
dimension at least 3 + 2r - 4 = 2r - 1. By a similiar argument W + (r - 2)gj
has dimension at least 4r - 5. Since r > 6 this contradicts our assumptionthat 2g£ has dimension 3r - 1. D
Corollary 1.1.9. Suppose that hrm and grn are two linear series such that grn + hrm
has dimension 3r, and both 2grn and 2hrm have dimension 3r - 1. Suppose,
moreover, that there exist linear series g\ and g\ so that hrm = (r - l)gxs + ¿P
and grn = (r- l)g| + S. Then gxT = gxs.
Proof. This follows from the argument given in (ii) above.
Lemma 1.1.10. If grn and g} are as in Lemma 1.1.8, then g„ + gj- has dimen-sion r + 2.
Proof. As in 1.1.7 our proof is split into two parts
(a) Suppose that grn - g¡. is simple, then write 2grn = [grn - gj.] + [grn + gj.].
If gn + gT has dimension r + 3,thenby 1.1.3 and 1.1.4, [grn-g¡-] + [grn + g\-]
has dimension at least r + 3 + 2(r - 2) = 3r - 1. Of course, it has exactly this
dimension but then by 1.1.4, since (r + 3) - (r - 2) = 5 > 3, gTn- gT must be
composite. This contradicts our assumption.
(b) Suppose that grn-g\ is composite, i.e., grn = (r- l)gj.+^'. Suppose that
grn + gj has dimension r + 3, then consider the linear series grn + (r- l)g\+¿P .
It would have dimension at least 3r+r-3 > 3r-1, which is a contradiction. D
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222 L. A. DONOHOE
We reconsider 1.1.8.
Theorem 1.1.11. Suppose that Xp is a Riemann surface admitting two linear
series, grn and hrm, with (for argument's sake) m > n so that both 2grn and
2h'm have dimension 3r - 1, and grn + hrm has dimension 3r. Then a gj which
imposes two conditions on grn also imposes two conditions on hrm .
Proof. Our proof splits in the usual way.(a) Suppose grn = (r- l)gj.+^ . By assumption hrm + grn has dimension 3r ;
however, if g\ imposes three or more conditions on hrm , then (r - l)gj. + Wm
has dimension at least r + 3(r - I) = 4r - 3 > 3r.
(b) Assume now that g'n - g\ is simple. Since 2hrm has dimension 3r - 1
there exists a linear series h}¡ so that hrm — hr~2s + hxs. We shall show that
/zj = gj..
Suppose that hrm - hxs is composite of dimension r - 2 and that hxs ̂ g\..
Then, there exists some ¿? so that hrm = (r- l)hxs+¿P . Now, hxs must impose
three or more conditions on grn . The linear series W — grn + (r - l)h$ has
dimension at least r + 3(r - I) = 3r + (r - 3). Since r > 3, W cannot be
contained inside a linear series of dimension 3r, which is a contradiction.We may suppose that hrm - hxs is simple and of dimension r - 2 .Then we
can write
grn + hrm = grnZ2T + hr-}s + [gl + hXs].
If Aj and g\. are distinct, then hxs + g\- has dimension at least three. Since
Kñ-s is simple, the linear series hr~2s + [g\. + hxs] has dimension at least r + 4.
By 1.1.4, the linear series grnZ2T + hr~}s + [g\- + h^] has dimension at least
r + 4 + 2(r - 2) = 3r. Of course, by hypothesis, it has exactly this dimension.
However, by 1.1.5, since r + 4-(r-2) = 6>3, grnZ2T must be composite,
which contradicts our assumption. G
Remark 1.1.12. We note that if grn is fixed point free and g\- imposes two
conditions on grn , so that the linear series grn - g\- is simple, then grn - g\ is
also fixed point free. If gr„ - gf = gr~2 + %, where % is effective, then grn
contains the linear series gr~2 + gj* which has dimension r. Therefore, grn
contains fixed points, which is a contradiction.
Corollary 1.1.13. Suppose that grn and hrm are two linear series satisfying the
assumptions of 1.1.11 and that gr„ - gj is simple. Then W = grn - g\ + hrm
has dimension 3r - 3.
Proof. Since gj. imposes two conditions on both grn and hrm , W has dimen-
sion at most 3r - 3 . If it has dimension 3r - 4, then since gr„ - g\- is simple
and fixed point free, we find, by 1.1.4, that hrm = [grn - g\-] + h}¡, for some AJ..
By 1.1.11, we must have g\- = h xs , a contradiction. D
Lemma 1.1.14 (Accola). Suppose that grn and hsm are two special linear series,
and that grn is simple. If grn + hsm = g£+n+t where t > 0 and r > s + 1 +1 then
hsm imposes no more than s + t + i conditions on grn .
Proof. Let & be the fixed point set of K-hsm . Choose a general divisor % of
hsm that corresponds to m distinct points in general position in Ps. We may
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GENERALISED CASTELNUOVO INEQUALITIES 223
assume that (&, 9~) - 0 and that
% = X\+ ■•• + X2s+t + X2J+f+i +-h Xm
where xx-\-h x2s+t imposes independent conditions on grn + hsm .There exists
a divisor 3 in hsm , so that (3, %) = xx A-Y xs-X. If 2 is any divisor
in grn containing xs H-h x2s+t — recall that s + t + 1 < r — then 2 also
contains the points x2s+t+x H-h xm . So 2 contains m - s + 1 points of F.
Since hsm is special, hsm imposes m - s conditions on K. Therefore, it cannot
impose more conditions on grn which, also being special, is a subseries of 'K.
Therefore, W c 2 . Thus hsm imposes at most s +1 + 1 conditions on grn . G
Corollary 1.1.15. Suppose that grn and hrm satisfy the hypotheses of 1.1.13.
Then hrm = grnz\ + g°m_n+T.
Proof. Note, as a consequence of 1.1.11 and 1.1.13, that hrm + gr~2T has
dimension exactly 3r-3. By 1.1.14, gr~2T imposes at most r conditions on
hrm , however it cannot impose fewer conditions since if it imposed only r - 1
conditions, there would be a gxm_n+T which imposed two conditions on hrm ,
which we have seen is impossible. Therefore
hm = gn-T + gm-n+T- ^
In 1.1.21 we shall extend this result to cover the situation where both grn - g\
and hrm - h\ are composite.
Remark 1.1.16. Suppose that our Riemann surface admits, in addition to a grn
for which 2grn has dimension 3r - 1, a simple gr~x so that the dimension of
gn + gmX equals 3r-2. Then by 1.1.3 g^"1 is a subsequence of grn , and
since gj imposes two conditions on grn it cannot impose more on gr~x.
When we begin to consider the consequences of Xp admitting several grn 's
attaining equality in a generalised Castelnuovo inequality we need some infor-
mation about the dimension of multiples of our g\-. Our next result provides
this information.
Lemma 1.1.17. Suppose that Xp admits j > 3 linear series of dimension r>6,
so that 2gi has dimension 3r - 1 for 1 < i < j, and gx-{-1- gj has minimal
dimension. Then the linear series 3gf, where g\- is the pencil imposing two
conditions on each g,, has dimension three, i.e., it is composite.
Proof. First, we note that if any of the g¡-gj is composite, then 1.1.6 assures
us that (r-l)gj. has dimension r-l, since r > 6 our result follows. Therefore,
suppose that g, - gj- is simple Wi. Consider the linear series W = g- + g2 + g-¡
and rewrite it as
W = [gx- gT] + [g2 - gT] + [g3 - gxT] + 3gT.
By hypothesis, W has dimension 6r, while Wx = W - 3gj- has dimension
6r - 12 + e . If 3gj. has dimension four, by applying Accola's method, we see
that W has dimension at least 6(r - 2) + 3 • 4 + 3 + 1 - 6r + 4, which is acontradiction. G
Remark 1.1.18. If we have two linear linear series, grn and hrm , such that 2g¿
and 2hrm have dimension 3r—1, grn + hrm has dimension 3r, and both grn-gj
and hrm - g\- are composite, then S and ¿P, as exhibited above, cannot be
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224 L. A. DONOHOE
empty. If this were the case, and m = n , then we would have grn = hrm, while
if m > n , hrm would have fixed points.
Theorem 1.1.19. Suppose that we have two linear series grn and hrm, both of
dimension r>6 and both simple. Suppose that 2grn + 2hrm has dimension lOr-
2, the minimum possible (this implies that 2grn and 2hrn both have dimension
3r - 1 and that 2grn + hrm and 2hrm + grn both have dimension 6r - 1). Let gj
be as usual, and suppose that at least one of grn - g\ or hrm - g\ is composite.
Then there exist linear series g and h both of dimension r + 2, both simple,
which satisfy the hypotheses of 1.1.11 (with r replaced everywhere by r + 2).
Proof, (a) Suppose that hrm - g\ is composite and that grnZ2T is simple. Let
hrm = (r - l)gj. + S. Let g = grn + g\. and h = hrm + gxT. Recall that both g
and h have dimension r + 2 by 1.1.10.(i) We show that 2g has dimension 3r + 5. To see this, first consider
the linear series g + g'n. We claim that it has dimension 3r + 2. If not,
then since 2grn has dimension 3r - 1, each time we add a multiple of g\
to 2grn, we increase the dimension by at least four. Rewrite 2grn + hrm as
g'n + gn + (r _ 1 )g\ + @ and consider the linear series grn + grn + (r - 1 )g\-. It has
dimension at least 3r - 1 + 4(r - 1) = 6r - 1 + (r - 4), which is a contradiction
because r > 6.Suppose that 2g has dimension 3r + 6 or more, then every time we add a
multiple of gj- to g+grn we increase the dimension by at least four. Rewrite the
linear series 2grn+hrm as g+g¿ + (r-2)gj.+¿f and then consider the linear series
g + gn + ir - 2)«?f • It has dimension at least 3r + 2 + 4(r - 2) = 6r - 1 + (r - 5),which is too big since r > 6 .
(ii) Using the fact that 2hrm + grn has dimension 6r - 1 and arguing in a
similar way, we can show that g + h has dimension 3(r + 2).
(iii) To show that 2h has dimension 3(r + 2) - 1 we proceed in stages
as before. Later (1.1.20) we shall need the fact that h + hrm has dimension
3r + 2, so we prove it here, along the way, so to speak. We use the fact that
W = 2hrm + grn has dimension 6r - 1. Rewriting W as h + h + gr~2T, we can
compute an upper bound on the number of conditions that grnZ2T imposes on
W. We rewrite W again this time as
(r - 2)gf + (r - 2)gj. + 2S + 3gf + grnZ2T
and using Accola's general position theorem we see that gr~2T imposes at least
2(r - 2) + 3 + (r - 3) + 1 = 3r - 3 conditions on W. This implies that h + hrmhas dimension no more than 3r + 2, which is what we wanted to show.
By assumption W = 2grn + 2hrm - 2gr~2T + 2h has dimension 10r-2 and we
shall show that 2grnZ2T imposes at least 6r - 7 conditions on W. We rewrite
W = 2grnz\ + (r - 2)gT + (r - 2)g| + 4gj- + M = grnz\ + Wx.
Then, by applying Accola's general position theorem, gr~2T imposes at least
2(r - 2) + 4 + 2(r - 3) + 1 = 4r - 5 conditions on W and so Wx has dimension
no more than 6r + 3. But gr~2T imposes at least (r-3) + 2r+l conditions on
Wx which implies that 2h has dimension no more than 6r+3-(3r-2) = 3r+5 .
Since it has at least this dimension we are done.
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GENERALISED CASTELNUOVO INEQUALITIES 225
(b) The case where both grn - g\. and hrm - gj are composite is easier, we
just mimic the argument given in (i) to complete the proof. G
Lemma 1.1.20. Let grn and hrm be as in 1.1.19. If grn- g\ is simple and
gn — gml + & > or if grn- g\ is composite and grn = (r - l)gj. + 3 then 3
lies in one divisor of g\-.
Proof, (a) Let us consider first the case g'n = gr~x + 3 where grn- gT = grnz\
is simple. By 1.1.12 gT~_}T is base point free. We saw in 1.1.6 that g^A-g^Z?
has dimension 3r - 4 however, by 1.1.3 g^~x + grnZ2T has dimension at least
3r - 5, since 3 ^ 0 we see that it has exactly this dimension. Since grnZ2T
is simple and fixed point free we can apply 1.1.4 to see that there exists aneffective divisor i* so that
Now we can write
grn = grn--2T + gT = gn--2T + Z+3
which implies that g} - 3 is effective.
(b) Now we consider the case grn = (r- l)g\- + 3 . We want to show that 3
imposes just one condition on g. Then g = g^+1 +3 where g£,+1 is simple
(because it contains grn ) then we use the fact that g + grn has dimension 3r + 2
to reduce the argument to the one given in the first case.
The linear series 2grn - grn + (r - l)gj- + 3 has dimension 3r - 1. However,
the linear series g„ + (r- l)gj. has dimension at least 3r-2 since g\- imposes
two conditions on grn , if it had dimension 3r — 1 then 3 would be a fixed
divisor of 2grn , which we know to be fixed pointfree. Therefore, 3 imposes
one condition on 2g¿ and therefore one condition on g which is a subseriesof2g£. G
Remark 1.1.21. Suppose that grn and hrm satisfy the hypotheses of 1.1.19 and
that both grn-g\- and hrm-h\- are composite. Then, by 1.1.7 and 1.1.11 there
are divisors 3 and F so that grn = (r - \)gxTA-3 and hrm = (r - \)g\ + % .
By 1.1.20 3 and <f are both contained in a single divisor of gj.. If ^ is a
fixed divisor contained in a single divisor g\ then we denote by ¿P the divisor
gj- ¿P . We may write
hrm = gr„ + gT-3 -W = [grn- gti+3 + i?,
which extends the result of 1.1.15.
1.2. This section contains the theorems which allow us to describe the natureof the relationship between two linear series on Xp , grn and hrm where grn and
hrm are such that 2g£ + 2hrm has minimal dimension. We suspect a relationship
similar to the one described in 1.1.21, our next result confirms our suspicions.
Theorem 1.2.1. Suppose that grn and hrm are two linear series on Xp so that
2grn + 2hrm has minimal dimension lOr - 2. Then there exist effective divisors
3,:, i = 1, 2 so that
"m = gn + gXT-3x-32,
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226 L. A. DONOHOE
and there are two possibilities:
(i) if (3\, 32) = 0 then 3X and 32 are fixed divisors which are contained
in different divisors of g\-.(ii) If (3X, 32) ^ 0 then 3X and 32 are contained in the same divisor of
gj-, but 3X + 32 is not contained in any divisor of g\-.
The first of these situations we shall describe as generic, the second we shall call
nongeneric.
Proof. Note that if both g'n - g\ and hrm - g\- are composite then the result
follows from 1.1.21. We may as well assume therefore that grn - g\ is sim-
ple.Consider the linear series hrm + g¡¡ - gT. We saw above, in 1.1.14, that it
has dimension 3r - 3, and that as a consequence of this
(1-1) hrm = grnZ2T + g°m_n+T.
Let us examine the linear series hrm + [grn + gj], we claim it has dimension
3r + 3 . If either grn - g\ or hrm - g\ is composite, then, using the arguments
of 1.1.19, we see immediately that g + h + gj has dimension 3(r + 2) + 3.
So we may assume that hrm - gj is simple. We establish first that 3r + 3 is a
lower bound. By 1.1.10 g„ + gj has dimension r + 2,by 1.1.3 [gn + gj-] + hrm
has dimension at least 3r + 2. If it has dimension exactly 3r + 2 then by1.1.4 there exists a gx_m+T so that grn + g\. = hrm + gx_m+T » Consequently
gJt-m+T imposes two conditions on the series grn ; if m > n this contradicts
1.1.8, since g\- has no fixed points. If n = m then by 1.1.8 we conclude that
grn = hrn also a contradiction. To see that 3r + 3 is also an upper bound, we
consider the linear series W - 2grn + hrm . By assumption, W has dimension
6r - 1. Rewrite it as [grn - gj] + [grn + hr„ + gf]. Now we rewrite it again, this
time as 2[grn - g\] + [hrm - g^] + 3g|. By Accola's count g'n - g\ imposes at
least 2(r - 3) + (r - 2) + 3 + 1 = 3r - 4 conditions on W. The claim follows
immediately.By 1.1.14, hrm imposes at most r + 2 conditions on grn + gj. By Accola's
general position theorem hrm imposes at least r + 1 conditions on grn + g\.
If it impose exactly r + 1 conditions, then Xp admits a g,J_m+r imposing
two conditions on both hrm and grn , which is impossible for the reasons given
above. Therefore hrm imposes two conditions on grn + g\- and there exists an
effective divisor g„_m+T so that
hm + gn-m+T = g'n + gT-
We also have
A« + gn-m+T = grn-T + 2gl
Therefore,
(1.2) ^ = g-2r + [2gj.-g„°_m+r].
Combining (1.1) and (1.2), we find that g°_„+r and g„_m+T are both con-
tained in 2gj. and that
2gj — g„-m+T + gm-n+T-
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GENERALISED CASTELNUOVO INEQUALITIES 227
Using an argument similar to the one given in 1.1.17 we can show that 2gj.
has dimension two. So we may write
2gj - gm-n+T = gn-m+T-
Clearly, neither gm_n+T nor g°_m+7- can lie in a single divisor of g\-. There-
fore, there exist divisors <§*■ and <£2 of degree 5 and q respectively, where
T + m- n = s + q, so that
g0n-m+T = [gT-^] + [gT-%2]-
Then, letting 3X= g\- %x and 32 = g\- - %2 , we obtain
(1.3) hrm = grn + gT-3x~32,
and either
(i) 3X and 32 lie in different divisors of gj,
(ii) 3X and 32 lie in the same divisor of g\-.
Note that if (ii) is the case, then we cannot have (3X, 32) = 0. If m > n
then g j. - 3X - 32 would be an effective divisor and hrm would have fixed
points, while if m = n , we would have hrn = grn. Both of these conclusions are
contradictory to our assumptions. G
Lemma 1.2.2. If we are in the generic situation above then (i) 3¡ imposes one
condition on grn, and (ii) grn-3¡ is fixed point free for each i.
Proof. If 3 is contained in a single divisor of gj, let us denote the divisor
gj-3 by 3. Let s and q be the degrees of 3X and 32, respectively. Let
us suppose that grn - g\. and hrm - gj are both simple, if this is not the case
then we can replace grn and hrm in (1.3) above by g and h, both of which
will then satisfy the hypotheses of 1.1.11, which are the only ones we really
need in this proof.(i) Suppose that 3X imposes two conditions on grn. Let g„Z2 = grn- 3X
and let grnz\ = gr„ - gT, by assumption the latter is simple and fixed point free.
Then
K = gn + gT-3x-32 = [gr„-3x] + [gT-32].
But, since 3X is contained in a single divisor of g\-,
grn-3x=grnZ2T + Wx=grnZ2.
Thereforegrn-} = [hrm - g}] +32 = hr~}T + 32
and we conclude that (3X ,32) ^ 0, which contradicts our assumption that
the 3i lie in different divisors of gf . Therefore, grn - 3¡ has dimension one
for each i.
(ii) Suppose that grn - 3X — gm~x +^ where //0, since gm~x contains
grnZ2T it is simple. By 1.1.20, (i), &+3X must lie in one divisor of g\-. Then
hrm = g£l+F + W2
again, by 1.1.20 we see that y + 32 must lie in one divisor of gj-, a contra-
diction. Note also that hrm - 3t is fixed point free and of dimension r - 1. a
We would like to make the same type of claim for the nongeneric case. Ofcourse, we cannot completely duplicate the last result, but the following lemma
introduces some structure into this situation.
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228 L. A. DONOHOE
Lemma 1.2.3. If grn and hrm are as in 1.2.1 and
Kn = gn + gT-3X-32 With (3X ,32)±0,
we shall prove that it is possible to find 3X and 32 so that 3x+32= 3X+ 32and
(i) (3X,32)=32.(ii) 3X imposes one condition on grn .
(iii) grn-3x has no fixed points.
(iv) (32,3~x) = 0.
Proof. Let us suppose that grn - g\ and hrm - gj- are both simple. If this is
not the case then we can replace grn and hrm in (1.3) above by g and h , both
of which will then satisfy the hypotheses of 1.1.11. The proof is constructive.
Suppose that
hm = grn + gT-3x-32
where (3X, 32) ^ 0 .
(i) Let & = (3X, 32). Then we can write Dx = sf + 3§ and 32 = W + 3§where (s/ , W) = 0 . Let us write
gT=s/+38 + W + i'.
Then we have
Kn = gn + gT - i& + & + W) - 38.
Clearly, if we let 3X = si + ¿8 + W and 32 - 33 , we have a pair of divisors
satisfying (i).
(ii) Next, we must show that 3X imposes only one condition on grn . Suppose
that it does not. Note that
hrm-gT = (g„-3x)-32.
Let W = gj - 3X. If 3X imposes two conditions on grn , then since gj- itself
imposes two conditions on grn , we must have
^-gj.+^2 = (g„r-gj.) + r.
Both sides have dimension r - 2 and I? is the fixed point set of the right-hand
side. Clearly, 32 and W have the same degree, but they cannot be equal, since
grn and hrm are distinct. Thus, we have a contradiction.
(iii) Suppose that grn - 3X - g[~x + È, where g\~x has dimension r - 1
and Ê is an effective divisor, possibly of degree zero. Assuming that lj¿0,
we proceed as follows, let g\ = si + 32 + W + 0? where 3X = s/ + 32 + W,
thenhrm = (gn_-3x) + (gx--32) = grl-x+sf+W + g + ê>
andgn = g[-x+3x+i.
Adding the expressions for grn and hrm , we obtain a linear series of dimension
3r. Since g\- = sf + 32 + W + %, we find that
gn +hrm = (g[-> + gT) + g[~x +(sf+W + 2Ê).
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GENERALISED CASTELNUOVO INEQUALITIES 229
Because grn - gj- is simple, it follows that g[~x is also simple, therefore the
linear series g[~x + gj- has dimension at least r + 1. By Accola's count, the
linear series (g[_1 + gj-) + g[~x has dimension at least 3r - 1. Clearly, its
dimension can be no greater than this — since Ê ^ 0, and grn + hrm is fixed
point free — so we may conclude that the divisor sf + W + 2Ê imposes onecondition on the linear series grn + hrm. Therefore, it imposes one condition on
grn and we may deduce that there exists some effective divisor y such that
sf + 32 + & + Ê = s/ + % + 2È +y.
So (32 ,Ê)-Ê and we may write
hrm = grn + gxT-(3x+È)-(32-È).
Note that (i), (ii), and (iii) are true for the divisors 3X — 3X + Ê and 32 -
32-i.
(iv) We may write hrm = grn + gj- - 3X - 32 where (i), (ii), and (iii) hold for
3X and 32. Let g[+1 = g[~x + gj- where grn = g[~x + 3X. By the argument
contained in (iii), g[+x has dimension r + 1. We have the relationships grn =
g[+x - Wx and hrm = g[+x - 32. Since grn and hrm are distinct, (32, ~3X) =
0. G
Remark 1.2.4. In any case where the relationship between two linear series is
nongeneric, we shall always assume that the pair of divisors involved enjoy the
properties of 3X and 32 described in the preceding lemma. Our next result
considers the consequences of Xp admitting three or more linear series which,taken pairwise, satisfy the hypotheses of 1.2.1.
Theorem 1.2.5. Suppose that Xp possesses three linear series g,, / = 1,2,3,
of the same dimension and degree. Let the dimension of each linear series be
r. Suppose that the linear series taken pairwise, satisfy the hypotheses of 1.2.1and the hypothesis that £)g. is of minimal dimension. Let gj- be the resulting
pencil imposing two conditions on each of the g,-. Let us further assume thatgi - gj- is simple for each i (cf. 1.1.19). We can write g2 = gx+gj--3i-32,
and g3 = gi+gj--g'x- .§2 for some 3X , 32 , %x, and g2 . Then {3X, 32}
n { Wx, %2} ¿ 0.
Proof. We have the two relationships
(D) g2 = gi + gT-3H-3>2
and
(E) g3 = gl+gT-%- %2
By the same type of argument used to produce (D) and (E), we can obtain
(F) g3 = g2 + gT-^ï-^2-
At this stage, there are eight possibilities depending upon whether or not (D),
(E), and (F) are generic or nongeneric. By symmetry, we are left with the six
cases:
(i) (D), (E), and (F) all generic.
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230 L. A. DONOHOE
(ii) (D), (E), and (F) nongeneric.(iii) (D) and (F) generic, (E) nongeneric.(iv) (D) and (E) generic, (F) nongeneric.(v) (D) generic, (E) and (F) nongeneric.
(vi) (D) and (E) nongeneric, (F) generic.
(i) From the three relationships, we find that
g3 = gX+2gT~3x-32-^x-^2
which implies that
gl = [gl-gT) + VgT-®l-®2-#i-$i].
However, we also have
ft = [ft - gT] + VgT -*i- «H-
Therefore
2gj- - r, - £T2 = 3gj. -3X -32 -9Ï -ft.
We may rewrite the last relationship as
(1.4) Wx+W2 = 3gj--3x-32-9x-92.
(a) Let us assume that 3\ , ft, and 3^2 each lie in a different divisor of
gj-. Then we can write
Wx+W2 = 3~x+ft + [gj-32-ft2].
Adding 32 to both sides we obtain
Wx+W2 + 32 = ~3x+ft + W2.
We have equality here instead of linear equivalence because, by 1.1.17, 3gj-
has dimension three and becausej^i , ft, and_y He^in different divisors of
gj-. We must have, therefore, I?» = 3X , or W2 = 3X, in either case we are
done.(b) Suppose that no three of the divisors 3X, 32, ft, ft are contained in
different divisors of gj-. Let us assume, for argument's sake, that ft and 3X
are contained in the same divisor of gj-, as are 32 and ft. If 3X + ft = gj-,
then from (1.4), we obtain
Wx+W2 = [gj--3X -ft] + ft + 3~2.
This implies that
gi+gi=W2+32.
We have equality here, not just linear equivalence, because <gx and %2 lie in
different divisors of gj-. However, this relationship implies that 32 and ft
are contained in different divisors of gj , which contradicts our assumption.
If (3X, ft ) ^ 0 , then since 3X and ft both impose one condition upon
g2 , we conclude that 3X = ft , i.e., 3X + ft = gj . However, we have already
disposed of this possibility.Finally, we must consider the case (3X, ft ) = 0 . Then, since 3X and ft
are contained in the same divisor of gj-, (3X, ft) = ft . Similarly, (ft, 32) =
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GENERALISED CASTELNUOVO INEQUALITIES 231
ft2 . Suppose that there exist divisors 3 and J? such that ftx + 3 — 3X and
ft2 + F = 32 . Then
g2 = gi + gj--3x-32,
g3 = gl + gT - %\ - %2 ■
ft = g2 + gj - ®\ - £¡2 + 3 + F.
Since g2 and g3 have the same degree, we must have 3 — F = 0. If this
is the case, and we add gx to the second expression for g3, then we find that
gi + g3 = 2g2 , counting dimensions leads us to a contradiction.
(ii) and (vi) (a) First of all, let us assume that 3¡ and F lie in different
divisors of gj-. Suppose that 3X lies in X, and Fi lies in &, then (%?, &) =
0 . We shall obtain a contradiction as follows. We have the relationship
g\+¥2 = 3gT-3x-32-ft-ft.
The left-hand side has dimension zero, since &[ + F2 is not contained in a single
divisor of gj-. Futhermore, it contains only points which lie in &, so adding
the divisor gT-32= 32 to it does not change the dimension since the result
may be written as 3gj. - F - fë2 - 32 , which clearly has dimension zero. Wehave
32 + gx+g2 = 3x+ft + ft.
Since (3~x,32) = 0, (¥X+W2,~3~X) ¿ 0, which is a contradiction. Thisargument is typical of the ones used below in (iii), (iv), and (v), and we shallnot reproduce it.
(b) Now we may assume that 3X and F lie in the same divisor of gj-.
We recall that T = deg3x + deg32 and that (3X ,32) = 32. Therefore,de%3x >T/2. Exactly the same is true for the pair of divisors If» and %2.
Suppose that deg^i > 7/2. We conclude that (3\,%\)±&. Since gx -3Xand gi -Fi are both fixed point free, and both have dimension r— 1, 3X = ^ .
If deg^i = 7/2, then 32 = 3X. If (3X, Si) = 0, then the restriction
on the degrees demands that ê\ = W2 = 3X. Adding g2 and g3 together,
we find that g2 + g3 -= 2gi. Counting dimensions gives a contradiction, so(3X, Wx ) ^ 0, in which case, we must have 3X = £[ .
(iii) We have (g\ ,%2) = %2. Suppose that &x is contained in %f, a divisor
of gj . Since 3X and 32 are in different divisors of gj-, they both cannot lieinside M?.
(a)_S_uppose that neither 3X nor 32 lies in %?. Then, as in (i) we find that
since Fi + i§2 has dimension zero
32 + Wx+W2=Wx+ft + ft.Since ft and ft are contained in different divisors of gj-, we must have, for
example, 32=ft, and therefore W\+£2=3\+ft which contradicts the factthat (3x,%f) = 0.
(b) Suppose that 3X lies in X, then 32 certaintly lies in a different divisor
and, once again, we have
32 + Wx+W2=3~x+ft + ft.
As before, we can reduce this relationship to
g\+g,2=&l +ft.
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232 L. A. DONOHOE
We note that since (F2, F) = F , (W2,ft2)^0. However, we have the two
relationships
ft = ft + ftl-Fi-F2 and g3 = g2 + gj- -ft -ft.
From 1.2.2 and 1.2.3, hfollows that_F2 and ft both impose one condition
on g3. Since both_g3 - F2 and g3 - ft2 are fixed point free, we conclude that
F = ft and that Wx =3~x, i.e., %x=3\.(iv) In this case, we may assume that 3X is not contained in the same divisor
of gj- as ft is. Again, we can write
32 + Wx+W2=Wx+ft + W2.
Since 32 and 3X lie in different divisors, and F_and F also lie in different
divisors of gj-, we must have 3X = Fi or Ü?i = F2.
(v) We shall see that this case is impossible.
(a) Suppose that neither 3X nor 32 is contained in the same divisor of gj-
as Fi and F2 are. Then, as before, the linear series 32A-W\A-W2 will have
dimension zero, and we find that
32 + Wx+W2=3~x+~ft+ft.
Since 3X and 32 lie in different divisors of gj-, we must have (F +F • -®i) 7^
0, which is a contradiction.
(b) Next, suppose that 3X lies in the same divisor of gj- as Fi and F do.
Once again, the divisor 32 + Fi + F2 will have dimension zero and we obtain
32 + ¥x+W2=3~x+ft + ft.
At this point, if we assume that F and ft lie in the same divisor of gj-,
we get a contradiction because 32 and 3X lie in different divisors of gj-.If Fi and ft lie in different divisors, then we obtain the relationships 32 =
ft +ft and 3X - F + F- Considering degrees, ^2 must have degree 7, acontradiction. G
Remark 1.2.6. If we have y > 5 linear series of the same degree and dimension,
and a linear combination of these linear series attains equality in the generalised
Castelnuovo inequality, then from 1.2.5 and the observation that if four or
more sets each contain two elements, and no two sets are disjoint, one element
is common to all of the sets, it follows that there exists a divisor 3 lying in a
single divisor of gj- so that, for 2 < i < j, g¡ = gx + gj--3 -3¡. Moreover,
since gi -3 is base point free (by 1.2.2 and 1.2.3), g, - gj- is base pointfree Vz, and
g,-gT = [grn-3]-3i
it follows that (3i, 3k) = 0 if i^k.However, if j = 4 and the relationships between the linear series are generic
then there may exist three distinct divisors 3 , F , and ft, all having the samedegree and all lying in different divisors of gj- so that
g2 = gi + gj- - 3f - F, g3 = gi + gj- - 3 - 9~,
and
g4 = gi + gf - F - y.
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GENERALISED CASTELNUOVO INEQUALITIES 233
We shall describe this situation as exceptional.
Remark 1.2.7. If Xp admits two linear series gi and g2, satisfying the condi-
tions of 1.2.1, then for some divisors 3X and 32, g2-gi + gj--3x-32. If(3X, 32) = 0, then when we map Xp to its image in Pr via gi, 3X and 32
will yield singularities which may fall together on the curve in Pr, however, by
assuming that the singularities are disjoint, we can give a lower bound on their
contribution to the delta invariant. In §3, we show that for generalised Castel-
nuovo curves, this estimate is sharp. If (3X, 32) = 32, then the singularities
on the image curve are not disjoint, one "lives inside the other." We can see this
in some sense by using the following rather heuristic argument. The larger divi-
sor 3X corresponds to a singularity S on the curve in Pr, the projection of the
curve into Pr_1 from the "point" S corresponds to the series gm~x = gx-3x.
Since g2 - 32 has dimension r - 2 the divisor 32 corresponds to a singu-
lar point on the curve in Pr_1. In any case the contribution of these nested
singularities to the delta-invariant is bounded below by the same lower-bound
as in the case (3X ,32) = 0. Therefore, as far as the calculations in §3 areconcerned, we may always assume that the relationship between any two linear
series is generic.
2
In this section, we break some ground in our classification of generalised
Castelnuovo curves. We perform the preliminary calculations that we will call
upon in §3 when we get down to the classification proper. We can use these
results to narrow our range of possible candidates, but alone they are not quite
enough to provide models of such curves. Allied to a little geometry, however,
they will allow us to complete the classification.
2.0.
Remark2.0.1. Given positive integers n, r, and j, consider n mod (jr- 1).
There exists an / e Z+ , so that if A- n-(jr-1)1, then 2 - (;' - 2)r < A < 2r.
We choose k , 0<k<j-l,so that
r + 1 <A < 2r if k = 0
2 <A < r if k = 1
2-(k- l)r<A< 1 -(k-2)r if2<k<j-l.
If k < 1 we can write n = (jl -k + 2)r-(l + t). If k = 0 then 0 < t < r - 1,while if k = 1 then 0<i<r-2.If/c> 2, we write n = (jl-k+2)r-(l-l+t)where 0 < t < r - 1.
Let us assume that we have a Riemann surface of genus p , Xp , which admits
j simple linear series of dimension r and degree n and that Xp is a generalised
Castelnuovo curve, by the preceding remark, given n, r, and j there are
essentially three possibilities:
(i) n = (jl + 2)r -(l + t) for some t, 0 < t < r - 1 (k = 0).(ii) n = (jl + \)r -(l + t) for some t, 0<t<r-2 (k = 1).
(iii) n = (jl - k + 2)r - (I + t - 1) for some k , 2 < k < j — 1 and t,0<f<r-l.
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234 L. A. DONOHOE
Perhaps some motivation for the preceding decomposition of n , the degree
of our linear series, is in order. Let Xp be a generalised Castelnuovo curve
admitting j linear series of degree n and dimension r, and let k, I, and t
be as above then
(i) If k = 0, and the linear series
W(j,0,l) = l[gl+g2 + --- + gj]
has smallest possible dimension, then its Clifford index is nonnegative,
while W(j, j - 1, / + 1) = W(j, 0, /) + gi will have negative Clifford
index and be nonspecial.1
(ii) If k > 1, and
W(j,k,l) = (l- l)[g, + --- + gk] + l[gk+x + ■ ■ ■ + gj]
is of smallest dimension then its Clifford index will be nonnegative,
while W(j, k - I, I) = W(j,k,l) + g\ will have negative Cliffordindex and be nonspecial.
In the following calculations, we find the conditions on n that
Wij, j - 1,1+ 1) and W(j, k, I) have negative Clifford index. We mustalso consider the possibility that either W(j, j — 1, / + 1) or W(j, k, I) is
exactly canonical, and has smallest dimension.
We recall that if gi • ■ ■ gj are j r-dimensional linear series, then Accola's
formula for the minimum dimension of lxgx H-h Ijgj, is given by
di) v(j,k,i)=±ík±^r-m^+j2¿:iiikr-í=l k=\ i<k
In [1], Accola derives 4*(/", k, I) and writes it in a form that enables oneto easily see that it is indeed a lower bound on the dimension of W(j, k, I).
In the following lemma, we express Accola's estimate for *¥(j, k, I) in a form
that allows us to quickly determine whether or not W(j, k, I) is nonspecial.
Lemma 2.0.2. Let Xp be Riemann surface admitting j linear series of dimen-
sion r so that
W(j ,k,l) = (l- l)[g, +-.. + gk] + l[gk+l +-+gj]l<k<j-l
or
rV(j,0,l) = l[gx+... + gj]
has the smallest possible dimension *¥(j, k, I). Then
yUtk,I) = UI-k + l)Ul-k)r Ul-2k)(l-l)_
Proof. Let
T(;- ; k f /} _ iß - * + Wl - k) r jjl - 2k)(l - 1)2 2
Fixing j and /, we proceed, by induction on k, to show that Q>(j, k, I) =
^¥(j, k, I). For k = j - 1 this assertion is easily verifiable and we omit the de-
tails. Let dk be the number of conditions that gk imposes on
1 Here, for our convenience we interchange the roles of #. and gj .
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GENERALISED CASTELNUOVO INEQUALITIES 235
W(j, k-l, I). From Accola's definition of *¥-(j, k, I) it follows automatically
that *¥(j,k-l,l)-6k = xi'(j,k,l). We note that 0k = (jl-k+ l)r-(l- 1)and leave the reader to verify that <P(;, k - 1, /) - dk = 0(;, k, I), whichcompletes the proof. G
Remark 2.0.3. Usually it is more convenient to write
^UJcJ)^Ul-k+'i)Ul-k)r (jl-k)(l-l) | fc(/-l)
n<(jl-k + 2)r-(l-l) +
2 2 2
If k > 1 and the Clifford index of W(j, k - 1, /) is negative, then
r(fc-!)(/-1)1. jl-k + l
and W(j, k - 1, /) is nonspecial by Clifford's theorem. We now indicate how
the decomposition of n mentioned earlier comes about. We exhibit the case
k = 0 and leave the verification for the cases k = 1 and 2 < k < j — 1 to thereader.
(i) (k = 0) Using Accola's formula to calculate the dimension of
W(j, j - 1, / + 1), we find it to be (for convenience we interchange the role of
gi and gj)
The degree of W(j, j-\ ,1 + 1) is (jl+l)n. Thus requiring W(j, j-l ,1+1)to have negative Clifford index, and so be nonspecial, means that
(jl +l)n< (jl + 1)(;7 + 2)r - (jl + 1)(/) + (; - 1)/.
Therefore,
. ;7+i . •The quantity in the brackets is always positive and less than 1, therefore
n < (jl + 2)r - I.
At the same time however, we must also have
n>(l+l)(r-l) + (j-\)lr+\ = (jl + \)r - I
in order to ensure that Castelnuovo's method may be applied. Taking n —
(jl + 2)r -(l + t) where 0 < t < r - 1 ensures that both of these conditions arefulfilled.
Remark 2.0.4. We note that (jl-k)n ¿ 2x¥(j, k, I) unless we take k = 0 andhave n = (jl + l)r - (I - I). However, this is just case (i) with t = r - 1. We
note that, generally speaking, if W(j, k, I) has negative Clifford index then we
cannot apply Accola's method to compute the dimension of W(j, k - 1, /),
the obstruction being the restriction on the degree. However, by the preced-
ing observation, if W(j,0,1) is exactly canonical and has smallest possible
dimension, then W(j, j - \,l + I) is nonspecial and has smallest possible
dimension.
Remark 2.0.5. Note that in the case k > 1 mentioned above we neccessarily
have / > 2. This implies that any pair of g,, for !</</, satisfy the
n < (jl + 2)r-l +
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236 L. A. DONOHOE
hypotheses of 1.2.1 namely that 2g, + 2gt has dimension 10r-2. This is not
the case if k — 0 and ¿~] g¡ is exactly canonical therefore whenever we consider
the case k — 0 we assume that / > 2.
Finally, the following lemma will be used repeatedly in the following sections.
From our work in § 1 we know that a generalised Castelnuovo curve will possess
a unique gj- imposing two conditions on each grn admitted by Xp . Our next
result, due to Accola in [1], gives us a lower bound on the value of 7. We
present the result in its full generality.
Lemma 2.0.6. If gj- is a complete linear series imposing two conditions on each
gn¡ f°r i < i < j where each grn\ is simple and
hg„\+--- + ljgnJj+g!n=K,
then T>YJi=lU + 2. If g£ is simple, then T > £j=1 lii + 3.
Proof. The argument presented here is culled directly from Accola [1]. Assume
that gj1 is composite, for our purposes we shall also assume that it is com-
pounded of gj-. Suppose now that 7 < Yj¡=\ h + i ■ Choose a divisor 3 in
gj-, away from the fixed points of gm , consisting of 7 distinct points. Write
3 = xx-\-YXr . Since all of the grn\ are simple and gj- imposes two condi-
tions on each of them we may choose 7 - 1 divisors (indexed in some fashion)
•®i £ gnk for some k e {I,... , j} so that (3, 3¡) = x¡ (such divisors exist
by Accola's general position theorem). Now consider the divisor S = Yl^i: • It
lies in K and contains 7-1 points of 3 . Since gj- imposes 7-1 conditionson K we see that S contains all of the points of 3 , which is a contradiction.
When g„\ is simple, then gj- imposes at least two conditions on it, and the
argument is essentially the same except that in addition to the 3¡ mentioned
above we can find a divisor 30 in g„\ so that (3q , 3) = x¡, for some i, then
we let Q-^o + E^. n
Remark 2.0.7. If Xp is a generalised Castelnuovo curve admitting j linear
series gx, ... , gj all of dimension r and degree n so that W(j, k + 1, /) isnonspecial while W(j, k, I) is special then, denoting the canonical series ofXp by K, the linear series
g{^ = K-W(j,k,l)
will be called the residual series. It may be simple or composite, and be base
point free or have base points. In the following section we shall assume that if
the residual series is simple then it is also base point free, we shall justify this
assumption in §3.
2.1. The case k = 0, / > 2 . Since W(j, j - 1, / + 1) = Wq is nonspecial, weproceed to use its degree and dimension to calculate the genus of our curve. We
find that
Since Wx = Wo- gx is special, for suitable m and p, we have
/[gl + • • • + gj] + g(n = K.
A direct calculation shows that m = (jl + 2)(r - t - 1). Calculating p is a
little more subtle, we assign to the degree and dimension of W0 the variables
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GENERALISED CASTELNUOVO INEQUALITIES 237
Ao and 80 > respectively. Next, we assign to the degree and dimension of Wx
the variables Ai and 61, respectively. We claim that
p = [Ao-Ax]-[e0-ex]-i.
From the Brill-Nöether form of the Riemann-Roch Theorem, m - 2p — Ax -
2©[. Since m = 2p - 2 - Ax and p = Ao - 60 , our claim follows. Clearly,A0 - Ai = n, so it remains to compute the value of 80 - ©i. However, this
quantity is exactly the number of conditions that gi imposes on W0. Since
everything is "tight,"2 it is equal to (;' - l)lr + (I + l)(r - 1) + I = jlr + r - I.Therefore, since n = (jl + 2)r - (I + t),
p = r - 1 - t.
We consider the linear series Lx - gx + g„z'~x and L2 = Wx - gx. SinceLx+L2 — K, Brill-Nöether implies that both series have the same Clifford index,and we can use this fact to calculate the dimension of Li. Let this quantity be
px .3 The degree of Lx equals 2« - (j- 1)1 - (jl + l)t-2, whereas the degree
of L2 is Ai - n and its dimension equals @x - 9X where 6X is the number of
conditions that gi imposes on Wx. We have
2« - (j - 1)/ - (jl +l)t-2-2px=[Ax- 28(] - [n - 26x].
By "tightness", 0, = (;' - l)(/)r + /(/•- 1) + 1 and we find that
px =3r-2t-2.
Remark 2.1.1. If t > 1 then g„\ is composite, if t > 2 it follows directly from1.1.4, while if t — 1 and gr~2 is simple then 1.1.4 implies that it imposes
r - 2 conditions on each of the g,, this in turn implies the existence of j
distinct g,J_m 's each imposing two conditions on g, Vz, this is impossible. If
gm is composite then, by 2.0.6 and 1.1.5 , it is compounded of a gjl+2 which
imposes two conditions on each g,.
Remark 2.1.2. If t = 0, then gr~x may be simple or composite. If gT~x is
simple, then Xp admits a gj- which imposes two conditions on each of the g,
and also on gr„zx, by 2.0.6, 7 > jl + 3 . If gr~x is base point free then 1.1.4
implies the existence of an effective divisor 3 so that
gi=g;-'+^.
If grñ ' has base points and if g„f}E is the moving part of the series we have
gi = gnT-e +-Sr- From 1.1.20 we see that the divisor 3 , above, is contained
in a single divisor of gj-.
Lemma 2.1.3. Suppose we are in the situation mentioned in 2.1.2 then by The-
orem 1.2.1 there exist divisors 3X and 32 so that
(2.2) ft = ^+<?l_3r1_^2.
(i) If the relationship between gx and g2 is generic, then either 3X — 3
or 32=3.(ii) If the relationship between the two is nongeneric, i.e., (3X, 32) = 32,
then 3 =3X.
2From now on we shall be dealing with the case where all of the previous inequalities in estimates
of the dimension are equalities. We shall refer to this as tightness or sharpness.
3Obviously g¡ + gü, has dimension px for i = \, ... , j .
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238 L. A. DONOHOE
Proof. We may suppose that grm ' is fixed point free. Substituting for gi in
(2.2) we find thatg2 = gr~x +3 + gj--3x-32.
The dimension of g2 + gr~x is p\ = 3r - 2. Therefore, there exists a divisor
F so that g2 = gr~x + F . It follows that
F = gj- + 3 -3x-32.
There are two possibilities to consider:
(i) If we are in the generic situation then 3 , 3X, and 32 cannot all lie in
the same divisor of gj-. Let us assume that 3X and 3 lie in distinct divisors
of gj-. Adding 32 to both sides of our equivalence relation gives
% + 32 = [gxT-3x]+3.
We have equality because the right-hand side has dimension zero. Noting that
32 and 3X lie in distinct divisors of gj-, we conclude that 3 = 32, and
F=WX.(ii) In the nongeneric situation we may suppose that (3X ,32) = 32, and
that gi - 3X has dimension r - 1 and is fixed point free, as is gx-3 . Then,
since 3X + 32 has dimension zero,
3\+32 = gT-%+3.
Since (3X, 3) ¿ 0, we conclude that 3X = 3 . O
2.2. The case k = 1. Suppose that n = (jl+ l)r -(l + t). W0 = l[gx +■ -- + gj]is nonspecial, so that we may use its degree and dimension to calculate the genus
of Xp , we find that
07+1)07) jljl+l) .P 2 2 J '
We write the canonical series as
(/-l)gi+/[g2 + --- + ft] + g£=K.
Let us denote by px and p2, respectively, the dimension of gi + g„\ and
gi + gin ■ for 2 < i < j. We may establish the following:
(i) m = (r-f-2)07+l) + 0'-l)/.(ii) p — r-t-2.
(iii) px = r + 2p.(iv) p2 - r + 2p + 1, for 2 < i < j.
Note that px has attained its mimimum possible value and we may apply 1.1.4
and 1.1.5 . There are two possibilities to consider if / = 0, and one otherwise.
If t = 0 we may be able to renumber our linear series so that gr~2 is simple,
this is the case we consider in (a) below. If t = 0 and no such renumbering is
possible or, if t > 1, then grm~'~2 must be composite and we analyse this case
in (b) and 2.2.1.(a) Suppose that t = 0, and gr~2 is simple. Noting that n-m = y7 + 2, we
find that Xp possesses a gjl+2 so that
gl = g'm2 + gjl+2.
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GENERALISED CASTELNUOVO INEQUALITIES 239
Furthermore, gxl+2 imposes two conditions on each g,.
(b) Suppose that g^''2 is composite. By 1.1.5 it compounded of a gj-
imposing two conditions on gx and gr„7l~2 = (r-t-2)gj- + 38 for some fixed
divisor 38 . By 2.0.6 7 = jl + e + 1 where e > 0. We may write
m = 07 + 1 + £)(r -2-t) + (j-l)l-(r-2- t)e,
and so 38 has degree (j - 1)1 - (r - 2 - f)E .We know that g2 = gi + g)i+e+x -3)x - 32 and that the sum of the degrees
of 3\ and 32 is jl + 1 + e. The following lemma will give us some more
information about the degrees of 3X and 32 .
Lemma 2.2.1. 38, the base locus of grm~'~2, contains either 3X or 32.
Proof, (i) First we remark that g\ + 38 has dimension 1. This is immediate
if t < r -2, otherwise (r - t - 2)gj- + 38 would have dimension greater than
r - t - 2. If t = r - 2, and gi - gj- is simple, then consider gx+ 38,'xt
has dimension r, rewriting it as [gi - gj-] + [gj- + 38], we see that gj- + 38cannot have dimension greater than one, if it did gi +38 would have dimension
r-2+4+E > r where e > 0. If g2-g\ is simple we can use a similar argument.
If gx = (r - l)g\ + 3° , and g2 = (r - l)gj- + €, then we use the fact that
gi + 38 has dimension r. Write gx + 38 = (r - 2)gj- + [gj- + 38] + 3d . Ifgj + 38 has dimension two or more, then gx+38 will have dimension at least
2 + 2(r-2) = 2r-2, which is a contradiction.
(ii) Let us assume that the relationship between gi and g2 is generic. From
the relationship
(/ - l)g, +l[g2 + g3 + •• • + ft] + (r-t- 2)g\ + 38x = K,
we obtain
(/ - l)ft + /[ft +g3 + — + gj] + (r-t- 2)gT + 382 = K
by interchanging the roles of gi and g2. Equating these two expressions gives
g2+mx =gx+&2.
We use the fact that
g2 = ft + gf -&i -3k = [ft - gT] + 3¡ + W2
to obtain
3x+32+38x=gj-+382.
38x cannot contain both 3X and 32, suppose that (3X ,38x) = 0. Then,
writing gj. = 32 + 32, we obtain
3[ + 38x =32+382.
We have equality because gj- + 382 has dimension one and 32 cannot be
contained in 382. For if it were, we would have 382 = ft+ 32 which would
imply that
3x+38x=g\ + ft,
i.e.,
38x =3x+ft
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240 L. A. DONOHOE
which contradicts our assumption that (3X, 38x) = 0 . So we have
3[ + 38x =32+382
which implies that 32 c 38x.(iii) Assume now that the relationship between gi and g2 nongeneric. By
assumption g2 = gx + g\ - 3 - 32 with (3, 32) = 32 . We want to showthat either (3, 38x) = 3 or (32, 38x) =32. The argument is the same as in
the generic case, until we arrive at the relationship
3+~32+38X = gj-+382.
If (382,W2)=W2, then clearly, (38x, 3) = 3 . If (382,3¡) ¿3¡, then
3 +33x =32+382.
Since (3, 32) = 0 , we must have (38x, 32) =32. G
Corollary 2.2.2. If we are not dealing with the exceptional case for j = 4, and
there is no reordering of our linear series which allows case (a) to happen, then
gi = gx + gj- - 3 - 3¡ for some divisor 3 . It follows from 2.2.1 and 1.2.6
that either 3 ç 38 or YJi=2 3iÇ38 .
2.3. The case k > 2. Suppose that n = (jl - k + 2)r - (I + t - 1) for some kwith 2 < k < j — 1 and 0 < t < r — 1. We saw before that
Wo = (l- \)[g2 + ••■ + &] + /[ft + gk+i + ■ ■ ■ + gj]
is nonspecial, and that Wx = W0 - gx is special. The genus p of our curve is
pmUI-k + WI-k + l)r _ow-»)_of_t + 1),.
The canonical series may be written
(/ - l)[gl +■•■ + &] + l[gk+\ +--- + gj] + gïn=K.
Let us denote by px and p2, respectively, the dimension of gi + g„\ and
gi + gm - for k + 1 < i < j . We may establish the following
(i) m = (r-i-l)07-/c + 2) + 0-Ä:)/.(ii) p = r-t-l.
(iii) px = r + 2p.(iv) p2 = r + 2p + 1 .
Remark 2.3.1. Note that there is nothing to distinguish g, from gi for l<i<k
as far as the calculation that gi + gm has dimension r + 2p is concerned. Note
also that px attains its minimum value and so we can apply 1.1.4 and 1.1.5 .In the light of (iv), above, we make the following claim which will be of use in
the case t = 0.
Proposition 2.3.2. If grn and gm~x are two fixed point free linear series so that
2grn has dimension 3r-\ and grn + gm~x has dimension 3r-l, then g^_1 g grn.
Proof. Suppose that grn = gm~x +F for some zero-dimensional divisor F. Add
grn to both sides. If grn + gm~x has dimension 3r - 1 , then F is a fixed divisor
of 2g£ . Since grn is fixed point free, this is impossible, a
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GENERALISED CASTELNUOVO INEQUALITIES 241
Corollary 2.3.3. If t — 0 in the situation above and gm is simple and fixed point
free, then by 2.3.2 gm = gr~x £ gk+i for l<i<j-k.
(a) If t = 0 and gm~x is simple and fixed point free (we briefly discuss the
possibility that gm~x, has fixed points in §3.3 ) we may apply 1.1.4 to see that
there exists an effective divisor 3, so that \3\ has dimension zero and
gx=gr~x+3.
By 1.1.20 3 is contained in some divisor of gj- where gj- imposes two
conditions on each of the g, 's and on gm~x. As we remarked above, g, + gm~x
has dimension 3r - 2 for 1 < z < k, and therefore there exist effective divisors
F so thatg/ = g;_1+F forl</<*.
The following result will eliminate a range of possible models.
Lemma 2.3.4. If j > 5 and k > 3 then gm~x cannot be simple.
Proof. We consider the case k = 3 < j - \ and j > 5. By the precedingremark, there exists an effective divisor 3 so that
gx=gr~x+3.
From 1.2.1
g2 = g\+gT-32-@2.
Substituting for gi in the last expression, we find that
g2 = g;-1+^ + gj--^2-^2,
since g2 = gm~x + F for some effective divisor F • We deduce, as in 2.1.2,
that 3 = 32 or 3 = S2. Suppose the former is the case. Arguing in a similarfashion with g3, there exists an effective divisor éf3 so that
g3 = ft + g\ - & - £3.
However, we also have
g4 = ft + gf --S>4 - Éf4 and g5 = gi +g\- 35-@s.
Consider our expressions for g2 and g4 . We know the divisors 3n, @n, 3, @\
cannot all be distinct. However, we cannot have either 3^ = 3 or (§^ = 3,
either of these would imply that gm~x C g4, which is false. So we must have
3n = S2 or St, = (S2. For argument's sake let us suppose that the former is
the case. By arguing in the same way with g3 and g4 we find that ^4 — &\.Therefore
g4 = ft+gf-^2-^3-
A similar argument for gs gives
ft = ft+g|-^2-^3.
Since g4 and gs are distinct, we have a contradiction. If k = j - I theargument given above does not go through but j > 5 implies k > 4 and so we
have g4 = gi + gj--3-Su with t#4 £ {(S2, S{\. However gm~x <£ gj implies
that ¿f4 e {<§2, Si} which is a contradiction. G
(b) If gm~'~x is composite, there exists a gx¡_k+2+E imposing two conditions
on gi, and gm~'~x =(r-t- l)gxt_k+E+2 +33 where 38 has degree 0 -k)l -
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242 L. A. DONOHOE
(r-t-l)e. Clearly, if t > 2, then gm must be composite. However, if t = 1,
then p = r - 2, and gm could conceivably be simple.
Suppose that i= 1 and that gm~2 is simple. By 2.3.1 we may apply 1.1.4
to gi and g2. As a consequence, one finds that there exist two distinct gx_m 's
which impose two conditions on gi, which is impossible. Therefore, we con-
clude that gm is always composite, possibly with base points.Therefore Xp
possesses a gj¡_k+e+2 imposing two conditions on each of the g, 's. We write
grm-t-l = (r-t-l)g}l_k+e+2+¿%,
where 38 has degree (j - k)l - (r - t - l)e. We have the following
Lemma 2.3.5. If gk+x = gi + gj¡_k+c+2 -31 -3k, then 38 contains either 3
or 3k-
Proof. The proof is exactly the same as in 2.2.1. G
Corollary 2.3.6. If we are not dealing with the exceptional case for j = 4, then
gi — gi + gj- - 3 - 3¡ for some divisor 3. We may combine 2.3.5, 1.2.6,and the fact that gi + gm~'~x has dimension 3r - 2t - 2 for 1 < i < k, to find
that Z¡=M3iC&.
Proof. Suppose that 38 =3 +ft. Then
g2 + (r - t - l)gj- + 38 = [g2 +3] + (r - t - \)gxT + ft.
However, since g2 + 3 has dimension4 r + 1, the linear series [g2 + 3] +
(r- t- l)g\+ft has dimension at least 3r- 2t - 1, which is a contradiction.
Note that this argument is valid regardless of whether the relationships between
the gi's are generic or nongeneric. G
3. The models
3.0. In this section, we derive all possible models for the generalised Castel-
nuovo curves discussed in §2. As we noted at the beginning of §1, if Xp admits
a single grn , with r > 6, so that 2grn has minimal dimension, then any of the
grn 's determines a birational map from Xp into a rational normal surface scroll
in Pr which will be denoted Y. If Xp admits j > 2 grn 's, where r > 6,
g\, ... , gj satisfying the hypotheses of §2 and if g, - gj- is composite Vz
then, since the hypotheses of 1.1.19 will be satisfied, we can always map Xp
into a scroll in Pr+2 using g, = g, + gj-, then g, - gj- will be simple, this fact
will allow us to perform our calculations as simply as possible. The referee hasremarked that instead of moving to Pr+2 to perform the calculations one could
note that a consequence of gi - gj- being composite is that the image of Xp
lies on a singular surface scroll in Pr, then one can desingularise and work with
the resulting curve.
We give a brief overview of smooth rational normal surface scrolls. Refer-
ences for the following material will be [8, pp. 522-527] and [2, pp. 95-100,113-123]. A rational normal surface scroll is a ruled surface of minimal degree
in Pr, whose general hyperplane sections are rational normal curves of degree
r - 1 . The Picard group of such a surface Pic Y = (H, L), where H and
4We see this because gj + 3¡ = [g\ - 3¡t\ + gj , and Si imposes only one condition on gx .
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GENERALISED CASTELNUOVO INEQUALITIES 243
L are the classes of a hyperplane and a line (of the ruling) in Y, respectively.
Furthermore, the intersections on Y are given by
(3.1) 7-7 = 0, L-H=l, H-H = r-l.
These relationships allow us to calculate Ky , the divisor class of the canonical
sheaf on Y. We know that KY = TH + ßL for some 7 and ß . If C is asmooth curve lying in Y, then from the adjunction formula, the genus, p(C),
of C is given by
(3.2) p(C) = 1 + C'C+2C'Ky.
Since the hyperplane sections H, and the line L, are smooth rational curves on
Y we can use this relationship twice to detemine that 7 = -2 and ß = r - 3.
Note that in the special case Y = P2, where H = L, we have the familiar
relationship Kp2 = -37.
Consider a linear series \TH + ßL\, suppose that C e \TH + ßL\. If C issmooth, by combining (3.1) and (3.2) we find that
(3.3) p(C) = m[^~1](r - 1) + [7 - l][ß - 1].
If C is of degree n then C • H = n gives ß = n - T(r - 1) and C • L = 7shows that C admits a pencil of degree 7 which imposes two conditions on
hyperplane sections. Therefore, given Xp a generalised Castelnuovo curve of
degree n with a gj- we can write down the divisor class of <p(Xp) in Pic Y
immediately it is TH+(n-T(r-l))L. If Xp is a generalised Castelnuovo curve
admitting j , j >2, linear series of the same degree and dimension we denote
its pencil by gj-. The linear series gi provides us with a map tp : Xp —> Ywhere Y is a rational normal surface scroll of degree r—\ in Pr. By Theorem
1.2.1 for 2 < i < /, there exist divisors 3¡ and S¡, imposing one condition
on gi so that g, = gi + gj- - 3i - Si, therefore the image of Xp under (p is
not smooth. We find that
(3.4) p<7JZ_z_n(r_1) + [r_1][n_r(r_i)_1]_^^-1),
where the sum runs over the singularities of (p(Xp) and the m, 's are the mul-
tiplicities of the singularities. Our task will be to find the class of, and the
singularities of, the image of our curve in Y. To do this, we need to knoweither the value of 7 or the multiplicity of one of the singular points on the
image curve. We can usually glean one of these pieces of information from the
residual series gm .Let
f(n,r,T)= T[T~ l](r- 1) + [7- l][n - T(r- 1) - 1]
it can be verified that f(n + T,r + 2, T) = f(n, r, 7) which reflects the fact
that we may work on a surface scroll in Pr+2 instead of in Pr. We now exhibit
the classes of the curves discussed in §2.
3.1. The case k = 0, / > 2. Recall that n = (jl + 2)r - (I + t) where0 < t < r - 1, that
/[gi + --+ft]+gr_1 = K,
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244 L. A. DONOHOE
where m = n - (j - 1)1 - (jl + l)t - 2, and that
We discuss the following three cases:
(a) t = 0 and gm~x simple.
(b) t — 0 and gm~x composite.
(c) t > 1 and gm~'~x composite.
The calculations involved in this section and the following ones are tedious but
elementary, we will provide the motivation behind each one and then present
the result.(a) Suppose that grm~x is simple and fixed point free, by 1.1.4
gx=grm-x+3
for some effective divisor 3 . Furthermore, by 2.1.3,
g,■■ = ft + gT - 31 - 3¿.
Since
l[gl+--+gj] + gml =K,
and gj- imposes two conditions upon each of the g, and also upon gm~x,
7 > jl + 3, by 2.0.6 . We shall see that jl + 3 is the only possible value for7. For the time being all we know is that
<p(Xp)e\TH + [n-T(r-l)]L\
for some 7. Xp admits a unique gj- which imposes two conditions upon gi ,
and therefore, upon g, for 2 < i < j . From (3.4), p < f(T), where
f(T) = TlT~l\r- 1) + [7 - 1][« - T(r- 1) - 1]
_ {j _ 1)[7-Q-l)/-2][7-0-l)/-3]
[Q-1)/ + 2][Q-!)/+!]2
We note that f(T) is quadratic in 7 and that the coefficient of 72 is
-(/• + j - 2)/2, which is negative, therefore f(T) has attains its maximum
value at some 7. We find that
t=(2;7 + 5)r-M2;2-4/)/ + 5/-82r + 2; - 4
Since
|;7 + 3-7|<I,
/07 + 3) is the maximum value of f(T) for 7 e Z. A futher calculation
shows that f(jl + 3) — p .Now we must investigate the possibility that gm~ ' has fixed points. We recall
Accola's lower bound on the dimension of the linear series
Wo = lXgrn\+--- + ljgn'1
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GENERALISED CASTELNUOVO INEQUALITIES 245
where each grn\ is simple and r¡ >rk if i < k then Wq has dimension at least
,,(r,,...,r;;/l,...,y = ¿<MÍ|±ii,_M^i)+¿:E;Art.¡=1 k i<k
Now suppose that gm~x = grmz}E +ftc° where e > 1, and grmz}e is fixed point
free then we have
l[gi+--- + gj] + gm--E+fte° = K.
Using Accola's estimate one can show that the series W = l[gx H-\-gj] + gmz}ehas dimension at least p— 1. Since e > 1 W has degree strictly less than 2p-2and we have a contradiction, therefore if gm~x is simple it can contain no fixed
points.
Therefore the only possible model for q>(Xp) c Y when gm~x is simple, is
one which admits a gj/+3, and which has j - 1 singular points of multiplicity
/ + 1, and one singular point of multiplicity (j - 1)1 + 2. The class of such a
curve in Pic Y is [jl + 3]H + [(j - 1)1 + 3 - r]L.(b) Let us suppose that gm~x is composite. Since m = (r - l)(jl + 2),
we deduce, from 1.1.5, that T - jl + 2. Thus, the class of (p(Xp) in Pic Y is[jl + 2]H + [(j - 1)1 + 2]7. Apart from one case when j = 4, we have seen
in § 1 that if i > 2 there exist divisors 3 and 3¡, of degree x and 7 - x
respectively, each lying in some divisor of gxl+2, so that g, = gi + gJl+2 -
3 - 3¡. Our aim now is to calculate all possible values of x. We shall
consider the exceptional case for j = 4 later. We have the inequality, p <
[jl + 2][jl+l]r/2 + f(x) where
/(x) = [;7 + i][0-D/ + i]-[j7 + 2f+1][jl + 2-x][jl+l-x] x[x-l]
2 2 '
The coefficient of x2 in f(x) is -j/2 so f(x) has a maximum at some
C/-1)
point x . We find that
(2j2-2j)l + 3j-2
2;
Since|0-l)/+l-x|<l/2,
f(x), for xeZ, attains its maximum value at x = (j - \)l + I. To finish off
one shows that
/(;/-/ + 1) = -[;/][/+ l]/2.
Thus,
P < fiJl -l+l) + [jl + 2][jl + l]r/2 = p,
and we have exhibited a model for Xp in Pr whose class in Pic Y is
[jl + 2]H + [(j - I) + 2]7. The model possesses j singular points, j - 1
of multiplicity / + 1, and one of multiplicity jl - I + 1.The exceptional case j — 4. In this case there exist divisors 3, F, and
ft, all of degree 2/ + 1, and all lying in different divisors of g\l+2, so that
g2 - ft +gxl+2-3-g, g3 = gi +g)M-3-ft, and g4 = gx +g)M-%-ft -
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246 L. A. DONOHOE
The existence of such a model for Xp is easily verifiable directly from (3.4).
The class of <p(Xp) in Pic Y is (4/ + 2)H + (31 + 2)7, and the curve has threesingularities of multiplicity 2/ + 1, no two of which are contained in the same
line of the ruling on Y.(c) Now we must consider possible models for the case n = (jl + 2)r —
(I + t) where 1 < t < r — 1. By 1.1.5, since gm~'~x is composite and m =
(r - t - 1)07 + 2), there exists a gj/+2 which imposes two conditions on each
gi. All we have to do is to produce the correct singularities and we are done.
The calculations go ahead as before. The only model is one whose image inPr has class (jl + 2)H + (jl - I - t + 2)7 in Pic Y, and which possesses j
singular points, j — 1 of which have multiplicity / + 1, and one of which has
multiplicity 0 - 1)/ + 1. If t = r-\ then we do not get any information about
the gj- from gm , however from 2.0.6 we see that T > jl + 2 and T > jl + 2
will be excluded by arguing as in 3.2.1 below.Similarly, in the exceptional case when j = 4, we obtain a curve whose class
in Pic Y is (4/ + 2)H + (31 + 2 - t)L with three singularities of multiplicity2/+1.
3.2. The case k = 1. In this case we have n = (jl + l)r - (I + t), where
0 < t < r - 2. We write K, the canonical series as
[/-l]ft+/[ft+-- + ft] + g;-i-2 = K,
where m = n - (jl + 2) - jit, and
p = [;7]Q7+l]r _[;/][/+l]_;/f
We consider the following three cases:
(a) t = 0, gr~2 is simple.
(b) t = 0, gr~2 is composite, and gr~2 = (r - 2)gjl+E+l +38, where 38
is the fixed point set of gm~2. 38 has degree (j - 1)1 - s(r - 2). We
shall show that e = 0 is the only possibility.(c) t > 1, and gr~2~l = (r-2- t)gjl+e+l +38 , where 38 is the fixed point
set of gm~2~'. 38 has degree (j - 1)1 - (r - t - 2)e . We shall showthat e = 0 is the only possibility.
(a) If gm~2 is fixed point free then there exists a gjl+2 so that gi = gm~2 +
gji+2 ■ Suppose that we are not in the exceptional case. There exists a divisor
3 , so that for 2 < i < j , g¿ = gx + gxl+2 - 3 - 3¡. Let 3 have degree x.
From (3.4), p < (jl + l)(jl)r/2 + f(x), where
[jl + 2107 + 1] [x][x-l]/(x) = L/7+l]L/7-/4
-0-1)
2L/7 + 2- x][jl +l-x]
2
We note, as before, that f(x) is quadratic in x with negative x2 coefficient,its maximum is attained at
* = ((2;2 - 2j)l + 3j - 2)/2j.
Since
|0-l)/+l-i| = |(2-;)/2;|<l/2,
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GENERALISED CASTELNUOVO INEQUALITIES 247
f(x) attains its maximum value over the integers at (j -1)1+1. A calculation
shows that
fijl-l+l) = -[;/][/+1]/2.
Therefore,[jl+l][jl] [;/][/+1]
P - 2 2 P
and so we have exhibited a model for Xp, whose image in Pr has class
07 + 2)H + (jl + 2 - I - r)L in Pic Y . Our model has j - 1 singularities ofmultiplicity / + 1, and one singular point of multiplicity jl - I + 1.
If gm~2 has base points then Xp admits a gh+2+E where e > 1. One can
show using the same argument as the one given above that no such model exists.
The exceptional case j = 4. If j = 4 then it is possible that there are divisors
3 , F, and ft, each of degree 2/ + 1 so that g2 = gi + g¡¡+2 -3 - F, g3 =
ft + &J/+2 -3 -ft, and g4 = gi + gxM+2 - F - ft. Once again, it can be verified
directly from (3.4) that there is such a model. Therefore, if j — 4 and grnZ2ji_2
is simple, we can exhibit a model for Xp which has class (4l+2)H+(3l+2-r)Lin Pic Y, and three singularities of multiplicity 2/ + 1.
(b) gm~2 = (r- 2)g1/+£+1 +38 , where 38 is the fixed point set of grm~2 . 38
has degree (j - 1)1 - e(r - 2). First, let us suppose that e = 0.
If j t¿ 4, there exists a divisor 3 of degree x, which lies in some divisor
of gjl+l so that, for 2 < i < j, g,= g, + g]/+1 -3 -3h From (3.4)
p<(jl)(jl+l)r/2 + f(x),
where
m = um _ „ _ Ui±mn _ y _ „[//+1-xiu»-*] _ Mfczil.
/(.x) is quadratic in x with negative leading coefficient. It assumes its maxi-
mum value at
* = 0-l)/+2-
which implies that f(jl -1) and f(jl -1 + 1 ) are the maximum values of f(x)when x is restricted to the integers. One shows that
nil -D = M -/ +1) = -[;/][/+ i]/2.Thus,
< L/7+11L/7] _ [//][/+1] =P - 2 2 P'
and so we have iwo distinct models for Xp . Both have class 07 + l)H +
(jl - I + 1)7 in Pic Y, however, one has j - 1 singular points of multiplicity
/ and one singular point of multiplicity (j' - 1)1 + 1, while the other has j - 1
singular points of multiplicity / + 1, and one singular point of multiplicity0-1)/.
(c) If t > 1, then grm ' 2 is composite and is compounded of a g)¡+E+x
(from 2.0.6 7 > jl + 1 ), and
p = [ji + wi]r_\jw+n_jlt
If we assume that e = 0, then the calculation in (b) above shows that there exist
two models for Xp , both models for Xp have class (jl + l)H + (jl-l+l-t)L
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248 L. A. DONOHOE
in Pic Y, one has ; — 1 singularities of multiplicity / and one singular point
of multiplicity (j - 1)1 + 1, the other has j - 1 singular points of multiplicity/ + 1 and one singular point of multiplicity (j - 1)1.
Lemma 3.2.1. The models derived in (b) and (c) are the only possible ones for
Xp, i.e., if K - [(/ - l)gi + /(g2 + • • • + gj)] is composite, then there exist no
models for Xp admitting a gß+£+x which imposes two conditions on each g,
with e > 1.
Proof.Part 1. Suppose that n = (jl+l)r-l, that e > 1, and that we are not in the
exceptional case for j = 4. Then for some divisor 3, of degree x, contained
in a divisor of gj¡+e+x we have g, = gi + g]/+e+i -3 -3¡, and the inequality
P < fi(x, e), where
n \ 07 + E+ l]L/7 + e], ,. ... „., . , ...f(x, e) = H-j^-i(r -l) + [jl + £][;/ - / - e(r - 1)]
[jl + e + 1 - x][jl + e-x] _ [x][x - 1]U ' 2 2
fx(x, e) attains its maximum value, as a function of x, at
. = (;-l)(2j/ + 2£+1)+1=(._1)f+ l_e¿J ¿ J
From (2.2.2), either 3 c 38 or Y,¡=23i c 38. Since 33 has degree0 - 1)/ - (r - 2)e, and the degree of 3¡ is jl + e + 1 - x, we have
x<x0 = (j-l)l-(r-2)s if 3 c 38,
or
x>xx = 0- l)/ + e+ 1 if^^c^.;=2
Note that these two values of x are, respectively, greater than and less than x .
Therefore, if we can show that /i(xn, e) and /i(xi, e) are both less than p
when e > 1, our proof will be complete. By observing that Xq < jl — I, and
that jl-l + E <xx we can solve the problem by showing that fx(jl-1, e) and
fx (jl - I + e + 1, e) are both less than p .Omitting the details, one finds that
/•/•i / x e(fi-l) (J-2)s2+jEf(jl-l,E)=p- 2 'r - z--^-J—
and
fx(jl-l + E+l,s)=p- 2 'r-E.
Clearly if e > 1 both quantities are smaller than p .Part 2. Finally, we must consider the case n - (jl + l)r - (I + t), where
1 < t < r — 2. Recall that we have
(/ - l)ft + /[ft + • • • + ft] + (r-t- 2)gx!+E+l +38 = K,
where 38 has degree (j — 1)1 - (r - 2 - t)s. We shall show that there is no
model for Xp if e > 1. Unless we are in the exceptional case for j = 4, there
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GENERALISED CASTELNUOVO INEQUALITIES 249
exists a divisor 3 of degree x, lying in some divisor of gj¡+E+x, so that for
2 < i < j
gi = ft + gjl+e+l - 3 - 3i.
As above, p < f2(x, e), where
f2(x,e)=Ul + e+Wl + e\r-l) + [jl + E][(j-l)l-E(r-l)-t]
[X][X-1] [Jl + E+1-X][jl + E-X]
= fx(x, E) - (jl + E)t.
As a function of x the left-hand side attains its maximum value at
Jc = 0-l)/ + e + l/2-e//.
However, since either 3 c 38 or J^3,■■ c 38 , by 2.1.4, either
x < xo - (j - 1)1 - (r - t - 2)e
or
0 - 1)(;7 + E + l-x)<(j-l)l-(r-t- 2)e.
The second inequality implies that
x>xi = (j- l)/ + e+l+<5
where á > 0. Once again, since
Xo < (j - 1)1 < X < (j - 1)1 + I + E < XX
it is enough to show that f2(jl - I, e) < p and f2(jl -l+l+e,e)<p. Weknow from our earlier calculations with fx(x, e) that both of these inequalitieshold.
The exceptional case j - 4. Xp admits a g4/+£+1 imposing two conditions
on each g, and 38 has degree 3l-(r-t-2)E . Suppose that there exist divisors
3, F, and ft, all of the same degree d > 2/, so that g2 - gx + g\¡+e+x -
3-g,g3 = gi +g4'/+£+1 -3 -ft, and g4 = g, + gj/+e+1 - F - ft. Since
g, + gm'-2 has dimension 3r - 2t - 3, for 2 < z < 4, we see by Lemma 2.2.1
that two of {3, F, <?"} lie in 38 . However, by counting degrees we find that
this is impossible, and we conclude that there is no model for Xp . G
3.3. The case k > 2. We have j linear series of dimension r and degreen . We have seen before that n = (jl - k + 2)r - (I + t - I), where k e Z,
2 < k < j - 1, and, 0 < t < r - 1 . Then, we can write
(/ - l)[ft +•■• + &] + /[ft+i +••• + £/] + g;-'"1 = K,
where m = n - (k - l)(l - I) - (jl - k + l)t - 2 and
, . [// - *+21U. - * + n,. t//iy - il _ ul _ k+ 1)t,
Aside from the exceptional case, which we shall discuss on an ad hoc basis, thefollowing are the possibilities:
(a) t-0 and gr~x is simple, with Xp admitting a gj- which imposes two
conditions on each g, and g„fx.
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250 L. A. DONOHOE
(b) t - 0 and grm x is composite and equal to (r- l)gx¡_k+2+e +38, where
38 has dimension zero and degree (j - k)l - (r - l)e . We shall see in
this section that e = 0 is the only permissible value for e . Note that
gji-k+e+2 imposes two conditions on each g,.
(c) t > 1 and grn'~x = (r-t - l)gX[_k+2+E + 38, where 38 has degree
0 - k)l - (r - t - 1)e . We shall see, as in (b), that e = 0 is the onlycase which can occur.
(a) First suppose that gr~x is simple and fixed point free. Recall, from
2.3.4, if j > 5 and k > 3, then g„fx cannot be simple. Suppose that j > 3,k = 2 and Xp admits a gj-. Then there exists a divisor 32 , of degree / + 1,
so that gi = g„zx + 32 and g2 = gx+ gj- - 3 - 32. From 2.3.4 we see thatfor 3<i<j
gi = g\+g\~3- 3i.
From (3.4) p < f(T), where
f(T) = m[^"1](r - 1) + [7 - l][n - T(r - 1) - 1]
,,- n [/+!][/] [7-/-l][7-/-2]U ' 2 2
f(T) assumes its maximum value at
f = (2jl+l)r + 2/2r.
We note that
|;7+l-7| = |i-I| <I
and conclude that f(T) attains its maximum over the integers at jl + 1. A
computation verifies that f(jl + 1) = p and we have a model for Xp in Pr,whose class in Pic Y is (jl+l)H+(jl-l-r+2)L. Ourmodelhas j—\ singular
points of multiplicity / + 1, and one singular point of multiplicity jl - I.
The exceptional case j — 4. Suppose that there are three divisors, 3 , F,and ft, all of the same degree, each lying in a different divisor of gj-, so that
g2 = ft+gf-^-F,g3 = gi+gI-^-^, andg4 = gi+gj.-F-^. Ifk = 2, then as we saw above, one of the divisors 3 or F , has degree / + 1.
But then, 7 = 2/ + 2 < 4/ + 1, so this case cannot arise.If k = 3, then 3 has degree 2/. Then, 7 = 4/ and our usual calculation
exhibits a model for Xp whose image in Pr has class 41H + (31 - r + l)L in
Pic Y which possesses three singular points of multiplicity 2/.
Now we briefly mention the case where the residual series, although simple,
possesses fixed points. Suppose that
(/ - i)[g, + • • • + gk] + i[gk+x + ••• + &] + g;-+'£ = k,
where g£,+£ = gr~x + ftE° where e > 1 and g'~x is simple and fixed point
free. One can show using Accola's arguments that, for k + 1 < i < j, theseries gi + grm~x has dimension 3r - 1 , therefore 2.3.3 and 2.2.4 apply and a
repitition of the calculations given above show that no such model can exist ife> 1.
(b) Let us suppose that e = 0. Leaving aside for one moment the com-
plications involved if j = 4, we know of the existence of a divisor 3 of
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GENERALISED CASTELNUOVO INEQUALITIES 251
degree x which imposes one condition on gi so that, for 2 < i < j, gi
ft + gji-k+2 -3 -3i. From (3.4), p < f(x, e) where
f(x,E)=[jl-k + 2^l-k + l\r-l) + [jl-k+l][jl-l-k + 2]
(j - \)[jl -k + 2- x][jl -k+l-x] [x][x - 1]
2 2f(x, e) attains its maximum value at
¿J ¿J
Since 2 < k < j - 1,
\jl _ k - I + 2 - jc| = |0 -2k + 2)/2j\ < 1/2.
Therefore, f(x, e) restricted to the integers attains its maximum value at(j -1)1-k + 2, one can verify that /07 -I -k + 2) = p. Thus, we can exhibita model for Xp whose image in Pr has class [jl -k + 2]H + [jl -I -k + 3]7in Pic Y. The image of the curve in Pr has j singular points, j - 1 of which
have multiplicity /, and one of which has multiplicity 0 - 1)1 -k + 2.The exceptional case j = 4. If k = 3, and e = 0, then jl - k + 2 is not
divisible by 2, and so the situation in (b) is the only possible one. However, if
k = 2 then we can exhibit a model for Xp whose image in Pr has class 41H +
(31 + 1)7 in Pic Y and which possesses three singular points of multiplicity
21.(c) Finally, we consider the case n = (jl - k + 2)r - (I + t - 1) where
1 < t < r — 1. In this case
(/ - l)[ft + • • • + gk] + /[ft+i + --- + gj] + (r-t- l)g}i-k+2+t + & = K.
If we assume that e = 0, the calculations performed in (b) show that there
is a model for Xp and we do not repeat them here. The class of the image of
Xp in Pic Y is 07 - k + 2)H + (jl - k - I + 3 - t)L. The singularities of theimage of Xp in Pr are the same as those exhibited in (b), namely j - I of
multiplicity /, and one of multiplicity 0 - 1)1 - k + 2.
Lemma 3.3.1. If K-[(l - l)(g, + •■• + gk) + l(gk+\ +■■• + gj)] is composite,then the models derived above are the only possible ones for Xp, i.e., there exist
no models for Xp admitting a gJ¡_k+2+E which imposes two conditions on each
gi with e > 1.
Proof. We shall argue in exactly the same fashion as before. Let us supposethat we are in the nonexceptional case, then there exists a divisor 3 , of degree
x, so that gi = gi + gji_k+E+2 -3 -3¡, 2 < i < j. We have the inequality
P < f(x, e), where
r, s [jl-k + £ + 2][jl-k+£+ 1], ,, r.. .f(x,e)= ^-^-lfr-i) + \ji-ic + e+l]
• [07 -k + 2)r-(l + t-l)-(jl-k + E + 2)(r - 1) - 1]
_ , . _ Ajl-k + E + 2- x][jl -k + E+l-x] [x][x - 1]
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252 L. A. DONOHOE
We find that f(x, e) achieves its maximum value, as a function of x , at
. _ p-l)(2j7-2/c + 2e + 3) + lX~ 2j
= (j-l)l-k + E + 2+kr-lr-e^U.J 2 j
Let xo = (j - 1)1 - k + e + 2 . Since e > 1 and 2 < k < j, which implies that
\k/j - 1/2| < 1/2, we may conclude that x < xo + 1/2.
Because k > 2, we know from 2.3.6 that 2J¿=jt+i A- C 38, where ^ has
degree (j — k)l - (r — t - l)e, therefore
0 - k)(jl -k + E + 2-x)<(j-k)l-(r-t-l)E.
The last inequality implies that x > (j - \)l-k + E + 2 + ô = xo + ô > [x],since ô > 0. There are two cases to consider:
(i) If 1 < t < r - 1, it will be enough to show that /(jcn, e) < p, V e > 1.(ii) If t = 0 it will be enough to show that f(xo + I, e) < p, V £ > 1.
One finds that
(i) f(xo ,e)=p-2—r - te '
(ii) f(x0 + l,E)=p-(j-k)- £(£~ V - e,
which completes our analysis of the nonexceptional cases.
The exceptional case j = 4. If a model exists and Xp possesses a g4/_fc+e+2,
then we know immediately that Xp must have divisors 3, F, and ft all of
the same degree, and lying in different divisors of g\t_k+l,+2, so that g2 = gi +
gli-k+e+2-®-*> ft = gi+gh-k+e+2-3f-^r, andg4 = gi+g\,_k+e+2-g-9-.If k = 2, then e = 2p and ft C38 (Since g2 + gm has dimension 3r-2t-2,neither 3 nor F can be contained in 38 ) where ^" is the fixed divisor of
the residual series. However, 38 has degree 2/ - (r - t - l)e, while 3 , F,and <?" all have degree 21 + p, which is a contradiction. If k = 3, then one
of F, ^ is contained in 38 , which has degree less than /. Counting degrees
leads to a contradiction, a
4. The plane models
We choose (generic) points {xx, ... , xr-2} on Xp , so that no two points lie
in the same divisor of gj-. Then consider the series
g„2_r+2 = ft - [Xl + • • • + Xr-2] .
Since gi is complete, we observe that g2_r+2 is also complete and setting
d = n - r + 2, gj provides us with a plane model, % c P2 for Xp via
the map it : Xp \-> % where n is the birational map composed of a series of
projections through the points xx, ... , xr-2. We note that Wd has at least
r - 2 singularities of multiplicity 7 - 1, since each time we project through x¡
the 7-1 other points in the divisor of gj- are mapped to one point in P2.
Since gj- imposes two conditions on gx, it imposes two conditions on g¿ , i.e.,
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GENERALISED CASTELNUOVO INEQUALITIES 253
8d = gT + ¿d-T>
which indicates the existence of a singular point of multiplicity d - T on fé¿ .
All of the models possessing a gj- in Pr which we exhibited in §3 are trans-
formed into plane models of degree d . These plane models possess, in addition
to the singular points exhibited in §3, r-2 singular points of multiplicity 7-1
and one singular point of multiplicity d - T. Let Rx, ... , Rr-2 be the r-2
singular points on Q of multiplicity 7 - 1, let Q be the singular point of
multiplicity d-T on Q, let D = n(3) and D¡ = n(3¡) where 3 and 3¡are as in §3.
In general then, the plane model for a generalized Castelnuovo curve has
degree d — n - r - 2 and j + r - 2 singular points5. We note that we can
eliminate r-2, (if r is even) or, r - 3 (if r is odd), of the singular points by
performing a succession of quadratic transformations centered at Q and twoof the Ä,'s.
These quadratic transformations will only eliminate the singularities at the
R¡ 's if they are in general position. If this is the case, and we assume that r
is even, and that j = 2, then we arrive at the same model by subtracting off
(r - 2)/2 multiples of gj- from grn . Then, we are left with a gj , which has
three singularities, Q which is of multiplicity d-T, P which is of multiplicity
5 and Dx which is of multiplicity T - s. Such a model may be constructed
by taking a homogeneous polynomial F(X ,Y, Z) of degree d in P2 andrequiring the requisite singular points to be at the points [1,0,0], [0, 1,0],and [0,0, 1].
If the R¡ are in general position, and if we assume that r is odd and that
y' = 2, then we arrive at a model by subtracting off (r - 3)/2 multiples of gj-
from grn to obtain gj,. If we subtract off one more point, say x, then we
have a gj which has four singular points, Q which has multiplicity d-T, P
which has multiplicity s, Dx which has multiplicity T - s and R which has
multiplicity 7 - 1. If we take a quadratic transformation centered at Q, Pand R then, since Q has degree d-T and R has degree 7-1, the singularityformerly at P will vanish. Now we can construct the curve in P2 as before.
For j > 3 linear series the existence of plane models for generalised Castel-nuovo curves is not quite so clear, since, in general it will involve constructing
a plane model of degree d with j singular points. We shall not discuss this
problem any further except to note that there are (unique) models for all of theRiemann surfaces discussed in §5.
4.1. A summary of the models. Tables 4.1.1 -4.1.4 summarise the classificationof models for generalised Castelnuovo curves performed in §3. Our notation
is as follows: p is the dimension of the residual linear series, 7 is the degree
of the gj ,and S is the multiplicity of the largest singular point on the image
curve in projective space. There are j - 1 other singular points of multiplicity
T - S (apart from the exceptional case j = 4 when there are three singularpoints of multiplicity 7/2 ).
5 Except when 7 = 4 and there are r - 2 + 3 singular points.
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254 L. A. DONOHOE
Table 4.1.1Case k = 0 : n = (jl + 2)r- (l + t), / > 2
ßr-t- 1
177+2jl + 3
U -1)1+10-1)/+ 2
"(Residual simple.)
Table 4.1.2Case k = 1 : « = (jl + l)r - (/ + t)
r-t-2 77+177+177+2
O-i)/0-i)/+i0-i)/+i
*(Residual simple.)
Table 4.1.3The case k > 2 & 7 > 5 : « = (7/ - fc + 2)r - (/ + Z - 1)
Value of k ßr- V 77+1 0 -1)/
>2 t- 1 jl-k + 2 (j-1)1-k + 2
*(Residual simple.)
Table 4.1.4The exceptional case 7 = 4
Value of k
0 r-t- 1 4/+ 2 2/+1t- 1 4/+ 2 2/+1
ir4/4/ 2/
"(Residual simple.
5.0. In this section we deal with equality in a Castelnuovo inequality when we
have two linear series of different degrees, but of the same dimension. There
are two natural questions to be asked here. The first one is, given a complete
grn on a Riemann surface Xp , then p has a maximum value px, when does
the existence of a second linear series call it hrn+p depress p below the value
px ? The second question which we might pose is, given that Xp admits a
second linear series hrn+p, and supposing that p attains its maximum value
under these circumstances, are there models of such curves, and if there are,
are they unique?
We give a partial answer to the first question. Suppose that Xp admits a grn
and that p attains the maximum value allowed by the Castelnuovo inequality
for one grn. Let K denote the canonical series on Xp. We recall that if
n = (l + 2)r-(l + t), where 1 < t < r - 1, then
lSn ^ S(r_i_i)(/+2) Ä-
X,p admits a gx+E+2 where e = 0 or 1 by 2.0.6. The presence of the linear
senes ^n+l+e ft ' &l+2+e-3° -> where 3° and S are two generic points
of Xp clearly does not depress the genus below the Castelnuovo lower bound.
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GENERALISED CASTELNUOVO INEQUALITIES 255
In the following sections we construct the answer to the second question and
complete the answer to the first question posed above. In the remainder of this
section we establish some notation and indicate the nature of our argument.
Let *F(c*, ß) equal the dimension of agrn + ßhrn+p .6 An elementary calcu-
lation shows that
V(a,ß)={a + ß+^a + ßK-{a + ß)ia2+ß-l)+aß.
Let A(a, ß) denote the degree of agrn + ßhrn+p. We note that A(a, ß) =
(a + ß)n + ßp, and that
V(a-l,ß+l) = y(a,ß) + [a-ß-l] and A(a- 1, ß + 1) = A(a, ß) + p,
and
x¥(a+l,ß-\) = x¥(a,ß) + [ß-a-l] and A(a+ 1, ß- 1) = A(a, ß) -p.
If agrn + ßhrn+p is nonspecial, then the quantity A(a, ß)-x¥(a, ß) corresponds
to the genus of the curve Xp assuming of course that agrn + ßhr„+p is complete.
Suppose that agrn + ßhrn+p is nonspecial and has minimal dimension, let
p(a, ß) = ^(a, ß) - A(a, ß). Then p(a, ß) is a quadratic function of a if
we regard a + ß as a fixed quantity. To locate the minimum value of p(a, ß)
— which will correspond to the genus of our curve — we only have to find thevalue of a so that
Note that
and that
p(a + 1, ß - 1) > p(a, ß) < p(a - 1, ß + 1).
p(a-l,ß+l)=p(a,ß) + [p + ß-a+l],
p(a+l,ß-l)=p(a,ß) + [a-ß-p+l].
Once we know that agrn + ßhrn+p is nonspecial, and we have calculated the
upper bound on p, we get down to the task of establishing whether or not there
is a model corresponding to Xp . As mentioned beforehand, the key to finding a
model lies in producing a suitable gj-, and two divisors on Xp , 3X and 32,so that
K+p = gn + gT-3X- 32.
We shall have
(a-l)gn + ßha+p + g*=K or agrn + (ß-l)hrn+p + gin = K,
and we can extract some information about gj-, or about 3X and 32, from
the residual series gw . Specifically, gm will yield either the value of 7, or
it will give us a bound on the degree of 3X. Once we have this information,
we turn to a quadratic polynomial in two variables, f(x, e), to finish the job.
Typically x will be the degree of 3X and 7 will be a function of e . f(x, e)
will have negative x2 and e2 coefficents, and we show the maximum value of
f(x, e) is equal to p , in which case we have a model, or less than p in whichcase there is no model.
6 We always assume that we can calculate this dimension by Accola's formula. Of course
we are, at the same time, making a tacit assumption about n, a, ß, p, and r, namely, that
n + p> ar + ß(r — 1) + 1 . These are precisely the conditions that must hold for Accola's count to
be valid.
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256 L. A. DONOHOE
5.1. Let n = (/ + 2)r - (l +1). Suppose that Xp , our curve, admits two linear
series of the same dimension but of different degrees, grn and hrn+p . Following
on our work in §2 we consider the following two possibilities for n :
Case A: n = (2/0 + 2)r - (l0 + t0), h > 2, where 0 < t0 < r - 1.Case B: n = (2/0 + l)r - (/0 + fo), where 0 < t0 < r - 2.
The two possibilities for n may be described by n = (2/0 + 1 + SA)r - (lo + to) ■Where 0<to<r-2 + ôA, and ôA = 1 if we are in Case A and 0 if we are in
Case B.If Xp admits two linear series of dimension r and degree n , and p attains
its maximum value in the Castelnuovo inequality arising from these linear series,then from §2 with 7 = 2,
(h + sA- i)gn + ioK + gk;2+âA-'o) = K.
Let the genus of Xp in this case be denoted by p2 . Given a curve Xp admitting
two linear series of the same dimension but of different degrees, grn and hrn+p ,
where p2 < p < px, let K denote the canonical series for Xp, then we seekintegers y, and t , such that
[/o + 7]grn + Uo - r]hrn+p + g£ = K
for some p and m .
There are four distinct cases to consider in both Case A and Case B they arethe following:
(i) p = 2v and v = X(2r - 1 ) + /0 - A, where 0 < A < r - 1 .(ii) p = 2v and v = k(2r - 1 ) + fo - A, where r < A < 2r - 2.
(iii) p = 2v + 1 and v = X(2r - 1 ) + to - A, where r < A < 2r - 2.(iv) p = 2v + 1 and v = k(2r - 1) +10 - A, where 0 < A < r - 1.
We shall indicate the argument for (i) in Case A. Afterwards, we tabulate theresults for the other seven cases.
Case (i). Xp admits a simple grn and hrn+2u where
n = (2l0 + 2)r - (l0 + t0), with v = X(2r - 1) + t0 - A , for 0 < A < r - 1.
1. Calculating p. First, we calculate the Clifford index of [/0 + v + X]grn +
[lo-v+X]hrn+2u. Let a = /0 + A + z/,and ß = lo + X-v . We denote by C(a, ß)
the Clifford index of agrn + ßhrn+p . We find that
C(a,ß) = (a + ß)(r-A-l).
In the case under consideration, (2/0 + 2l)(r - A - 1) is nonnegative. So we do
not yet have an estimate for p . Next, we seek C(a + 1, ß). One can derive
the formulae
C(a+l, ß)-C(a, ß) = n- (2a + 2ß + 2)r + 2a
and
C(a, ß + l)-C(a,ß) = n + p-(2a + 2ß + 2)r + 2ß.
Using the first result we find that
C(a + 1, ß) = -(2/0 + 2A + 1)A - (/0 + A - i/)
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GENERALISED CASTELNUOVO INEQUALITIES 257
which is negative, therefore the linear series W0 = (a + l)grn + ßhrn+2v is non-
special. Therefore, we may use its dimension and degree to calculate the genus
of our curve7. The degree of W0 is (a + ß + l)n + 2vß, while the dimension
is (a + ß + l)(a + ß + 2)r/2 -(a + ß+l)(a + ß)/2 + (a+l)ß.We find that
p = (2/0 + 21 + 2)p/„-2. + 1)r + /l2_/a(;o+1)
-1/(1/ - 2/0 - 2A - 1) - (2/0 + 2A + l)/0.
Thus, for suitable p , and m we have
[/0 + V + X]grn + [lo-V + Wn+2v +gn\=K-
2. The residual series. As before, we calculate p, the dimension of the
residual series from the formula p = n - do - 1, where 0O is the number of
conditions that grn imposes on W0 . By sharpness, 0O = (2/o + 2A + l)r - (/0 +
X + u + l) + l, we find that p = r-A-l. Since 0<A<r- 1, 0<p <r- 1.A calculation shows that
w = (2/0 + 2A + 2)(r-A-l).
Next, we calculate the dimension, px, of grn + gm- Using the formula derived
in §2, /¿i = n + p- 6X, where 6X is the number of conditions that grn imposes
upon K - gm- Since everything is sharp in the formula for the dimension of
the preceding linear series, we find that 6X = (2/0 + 2A)r - (/o + A + v) + 1.
Substituting this value, and the expression for n , into our formula for px we
find that
(5.1) px - r + 2p.
Finally, we compute the dimension, p2 , of hrn+2v + gm . It equals n+p+2v-62
where 62 is the number of conditions that hrn+2l/ imposes upon K - gm . By
sharpness, 92 = (2/0 + 2A)r - (/0 + A - v) + 1, which gives
(5.2) p2 = r + 2p.
3. The models. From 1.1.4 we see that if p < r - 3 then gm is composite,
if p = r-2 or p = r-l then gm may be composite or simple. We investigate
all of the possible cases below.
(a) Suppose that gm is composite. Then, gm is composed of a g2\ +2X+2
which imposes two conditions on grn . From 1.2.1
K+2, = gn + ft!/„+2A+2 -3X-32,
where 3X + 32 has degree 2/o + 2A - 2z^ + 2. Since gm has no fixed points,
one of 3X and 32 must have degree at least /o + A - v + 1. Making use of our
results from §3 we use grn to provide a map <p : Xp -> Y where Y is a rational
normal surface scroll in Pr. Recall that if C e \TH + ßL\ is smooth then
),(C)=nr-ii(r-i)+[7._w_11
In the situation under consideration, the image of Xp under <p will not besmooth since both 3X and 32 impose one condition on grn. Let x be the
7 p(a + 2, ß- \)=p(a, ß) + [2v- 2v + 2] and p(a, ß + 1) = p(a, ß) + [2u-2v]
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258 L. A. DONOHOE
degree of 3X. Then we have p < f(x) where
f{x) = (2/o + 2A + 2)(2/0 + 2A+l)(r _
+ (2/0 + 2A + 1)(« - (2/0 + 2X + 2)(r - 1) - 1)
x(x - 1) (2/p + 2X + 2 - 2v - x)(2lp + 21 + 1 - 2v - x)
2 2f(x) attains its maximum at x = /0 + A-i/ + l. Evaluating, one finds that
f(x) — p. We see, therefore, that 3X and 32 have the same degree. A plane
model for Xp can now be produced by projecting into P2 and the resulting
curve (possibly after some quadratic tranformations, cf. §4) will have three
ordinary singular points.
(b) We see from (5.1) and (5.2) that gm cannot be simple if p = r -2,if
it were, then Xp would admit a gx_m and a gx+2l/_m , both of which impose
two conditions on grn , which is impossible.
(c) If p = r - 1, and gm is simple, then Xp admits a g\¡ +2x+E+2 ■ There
exists a divisor 3 of degree n - m imposing one condition on grn so that
gn = ft«-1 +3. Since p2 = 3r- 2, we can deduce by the same argument as we
used in 2.1.3 that hrn+2v = grn + gj- - 3 - 3X. Now, n-m = lo + X-v + 2
and so, p < /(e), where /(e) is the usual quadratic expression (cf. (3.4)). In
this case /(e) simplifies out to
, r 2 r+2p-i--ie2 + —e-
We conclude that if e = 1, there is a model for Xp in Pr admitting a g2l +2J+3,
and possessing two singular points of multiplicity ¡o+A+v + 2 and /o + A - v +1.This concludes our analysis of the the first case.
5.2. The calculations performed in the first case are typical of the ones for theother cases. The only subtlety required is when we have to deal with the cases
p = r - 1 or p = r - 2, and the residual series may be simple or composite.
In every case Xp admits a model where gm is composite. When gm is simple
there may not be a model. In these cases, calculating p2 (the dimension of
K+p + gm ) is usually the key to deciding whether or not a model may exist8. In
Tables 5.2.1 and 5.2.2 we have marked with an asterisk the entries corresponding
to a simple residual. The reader will notice, that in all but one case, it is possible
to have a simple residual. Now we tabulate the result of our calculations for all
possible cases. In each case the models exhibited can be (and have been) shown
to be unique. In our table we use the following notation:
(i) v = X(2r - 1) + io - A where 0 < A < r - 1 or r < A < 2r - 2.(ii) a and ß will be the integers which are functions of p such that agrn +
ßK+p wm have negative Clifford index. Once we have establised a and ß we
can obtain an upper bound on p .
(iii) p will be the dimension of the linear series gm suchthat agrn + ßhrn+p +
gm = K where {â, ß} = {a - 1, ß} or {â, ß} = {a, ß - 1}.
(iv) 7 will be the degree of the unique gj imposing two conditions on grn
and hrn+p T = a + ß+l or a + ß + 2.
(v) 5 = max{deg^i, deg^2} where hrn+p = grn + gj- - 3X -32.
8 For example in (b) above, since p.2 = r + 2fi, grm~2 could not be simple.
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GENERALISED CASTELNUOVO INEQUALITIES
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260 L. A. DONOHOE
For a given n and r we are in Case A or B. Given p, we find a and ß so
that agrn + ßhrn+p is nonspecial. Then we can calculate p . Next, we find p .
Once this is determined we can find 7, and the degrees of 3X and 32 are
determined since they can differ by at most 1 and add up to T - p. Finally,
the genus Xp of our Riemann surface can be calculated from the formula
P = T[T~l\r - 1) + [7 - l][n - T(r - 1) - 1]
S[S-l] [T-S-p][T-S-p-l]
2 2
and the result of this calculation agrees with the upper bound in (ii). We also
note that the arguments in §4 show that plane models for such curves can be
constructed.
Remark 5.2.3. To see that / (or / + 1 ) is an upper bound on p one just has to
examine the tables. Setting ß = 0 or ß = 1 (depending on how we calculate
p ) will leave us in the case of one grn . and give p = a = I or p = a = I + I.
We sum up our findings in this section in the following theorem.
Theorem 5.2.4. If r > 6 and I > 4 and n = (I + 2)r - (I + t). Let px be theCastelnuovo upper bound for a curve admitting one grn . There exist generalised
Castelnuovo curves of genus p <px admitting a grn and hrn+p Wp, 1 < p < I.
If t = 1 then there exists a generalised Castelnuovo curve of genus px admitting
a K+m ■
References
1. R. D. M. Accola, On Castelnuovo's inequality for algebraic curves, Part 1, Trans. Amer.
Math. Soc. 251 (1979), 357-373.
2. E. Arbarello, M. Cornalba, P. Griffiths, and J. Harris, Geometry of algebraic curves, vol. 1,
Springer-Verlag, Berlin and New York, 1985.
3. G. Castelnuovo, Sui multiple di una serie di grupa di punti appartenente ad una curve
algebraica, Atti Accad. Sei. Torino 28 (Memorie scelte) (1893).
4. J. L. Coolidge, A treatise on algebraic plane curves, Oxford Univ. Press, 1931; Dover, 1959.
5. M. Coppens, On G. Marten's characterization of smooth plane curves, Bull. London Math.
Soc. 20(1988), 217-220.
6. _, Smooth curves possessing many linear systems g\ , Arch. Math. (Basel) 52 (1989).
7. M. Coppens and T. Kato, Pencils on smooth curves implying special plane models (to ap-
pear).
8. P. Griffiths and J. Harris, Principles of algebraic geometry, Wiley, New York, 1978.
9. G. Martens, Eine Charakterisierung glatter ebener Kurven, Arch, der Math. 41 (1983),
37-43.
10. R. J. Walker, Algebraic curves, Princeton Univ. Press, 1950; Dover, 1962.
Kevin Street College of Technology, Kevin Street, Dublin, Ireland
E-mail address : 1. donohoeQdit. ie
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