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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 , admittingj linear series of dimension r and degree n Accola derived a polynomialfunction f(j, n, r) so that p < f{j, n,r) and exhibited plane models ofRiemann surfaces attaining equality in the inequality. In this paper we providea 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 commondimension but different degrees.

Introduction

In his paper [3] Castelnuovo produced a bound on the genus of an algebraiccurve 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 curveswhich possessed maximal genus. In his paper [1] Accola generalised the ideaby 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 anupper bound on the genus of the curve in terms of the number of linear seriesand 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 ofmaximal genus. We shall call curves of maximal genus generalised Castelnuovocurves. Martens [9], Coppens [5, 6], and Coppens and Kato [7] have consideredthe problem for the case r = 1. Coppens gives a classification of Castelnuovocurves possessing several gj 's in [6]. In this paper we consider the problem forthe case r > 6.

In this section we introduce the arguments used to produce the respectiveupper bounds and establish some notation and fundamental ideas. All of theseare 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. In1.2.1 we prove that if grn and hr are two such linear series then

K = g

218 L. A. DONOHOE

Pr lies on a rational normal surface scroll, the g\- corresponds to the rulingon 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 thecurves in 4. Section 5 investigates the situation when Xp admits grn and hrn+pand has maximal genus.

Definition 0.0. If grn and gsm are two complete linear series on a Riemannsurface Xp , with s 1 + Yj=\ U tne generalposition theorem tells us that grn\ imposes at least 1 + ]_. i, conditions ongff1 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 seriesrVo = hgr\ + --- + ljgni

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GENERALISED CASTELNUOVO INEQUALITIES 219

has dimension at least

**,....r,././,) = >|ilr,-M|zil + I:E;,V,7=1 k i 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 argumentto 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 surfacepossessing one grn , it was critical for him that the dimension of 2grn equal3r - 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 arational normal surface scroll FcPr [8]. If r = 5, then the curve may lieon the Veronese surface in P5. Moreover, Castelnuovo was able to deduce theexistence of a g\ on the curve cut out by the rulings of the scroll. Finally, sincethe 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 andhrm, and r > 6 such that W 2grn + hrm has dimension 6r - 1, the minimalpossible, then Xp possesses a gj- imposing two conditions on grn .Proof. Since W has minimal dimension it follows that 2grn has minimal di-mension and the result follows from Castelnuovo's observation. D

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220 L. A. DONOHOE

We quote, without proof, three results due to Accola [1] that we shall useextensively 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 ofgrn + 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 thedimension of grn + gsm is exactly r + 2s, then there are precisely three possibilitiesfor 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 = grx + gn_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 finalresult we quote is the following .

Lemma 1.1.5. Suppose grn is simple and without fixed points, gsm is compositeand 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 thefollowing two lemmas, we use them to describe completely the nature of grnwhen 2grn has dimension 3r - 1 and grn - g\ is composite.

Proposition 1.1.6. Suppose that Xp admits a complete, simple linear series grnand that 2grn has dimension 3r - 1. Let g\. be the linear series that imposestwo conditions on grn . Then grn + [gn - gf] has dimension 3r - 4.

Proof. Note that g} imposes at least three conditions upon 2grn and that byassumption, 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 ongrn), 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 leastthree. Therefore, rewriting W = (r - 2)gxs + g\ as (r - 3)gj + (g\ + gxs) andusing the fact that g\ imposes at least two conditions on g\ + g^ we see thatW has dimension at least 3 + 2(r - 3) = 2r - 3 . Since r > 6, this number isgreater than r which is a contradiction. Therefore, gj-gg- U

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GENERALISED CASTELNUOVO INEQUALITIES 221

If Xp admits a grn so that 2grn has dimension 3r - 1, then its image inPr lies on a rational normal surface scroll Y. The model will be determinedin part by the degree of the gj which imposes two conditions upon the linearsystem 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 grnand consequently, possess many distinct models. However, using an elementaryargument, we can show that there is only one such pencil.Lemma 1.1.8. // Xp admits a complete, simple gr so that 2grn has dimension3r - 1, then the gf imposing two conditions upon grn is unique.Proof. Suppose that Xp admits two distinct pencils, g\ and gj, with bothimposing 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'ssake 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, 2grnhas 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 1.1.3again, 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 existdivisors & and S so that grn = (r - l)gj. +&> and g'n = (r - l)g\ +f. Weconsider

2grn = (gs + gxT) + (r- 2)gx + (r- 2)gxT + &+.The linear series g\- + gj has dimension at least three and both gj and gjimpose at least two conditions on it therefore, W = (gj + gf) + (r - 2)gj hasdimension at least 3 + 2r - 4 = 2r - 1. By a similiar argument W + (r - 2)gjhas 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 + hrmhas 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 + Pand grn = (r- l)g| + S. Then gxT = gxs.Proof. This follows from the arg