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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


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


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



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


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



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


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.



(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,


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-



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.


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).



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.


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) =



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.


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.



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.


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 .



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 +


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



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 .


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.



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


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



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 -


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 .



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,


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



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 -


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,



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


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



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.


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



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]


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.,



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.


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.



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.


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/)



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]


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.


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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