Divisors on real curves

(de Gruyter 2003

Divisors on real curves

Jean-Philippe Monnier*

(Communicated by C. Scheiderer)

Abstract.LetX be a smooth projective curve overR. In the first part, we study e¤ective divi- sors onXwith totally real or totally complex support. We give some numerical conditions for a linear system to contain such a divisor. In the second part, we describe the special linear sys- tems on a real hyperelliptic curve and prove a new Cli¤ord inequality for such curves. Finally, we study the existence of complete linear systems of small degrees and dimensionron a real curve.

2000 Mathematics Subject Classification. 14C20, 14H51, 14P25, 14P99

Introduction

In this note, a real algebraic curveXis a smooth proper geometrically integral scheme overRof dimension 1. A closed pointPofXwill be called a real point if the residue field at P is R, and a non-real point if the residue field at P is C. The set of real points, XðRÞ, will always be assumed to be non-empty. It decomposes into finitely many connected components, whose number will be denoted bys. By Harnack’s the- orem we know that 1cscgþ1, wheregis the genus ofX. A curve withgþ1k real connected components is called anðMkÞ-curve.

The group DivðXÞof divisors onXis the free abelian group generated by the closed points ofX. LetDADivðXÞbe an e¤ective divisor. We may writeD¼DrþDc, in a unique way, such thatDrandDcare e¤ective and with real, respectively non-real, support. We callD_r (resp.D_c) the real (resp. non-real) part of D. In the sequel, we will say thatDis totally real (resp. non-real), ifD¼D_r(resp.D¼D_c).

By RðXÞ, we denote the function field ofX. Let PicðXÞdenote the Picard group of X, which is the quotient of DivðXÞ by the subgroup of principal divisors, i.e.

divisors of elements in RðXÞ. Since a principal divisor has an even degree on each connected component of XðRÞ ([4] Lemma 4.1), we may introduce the following invariants ofX:

* Partially supported by the EC contract HPRN-CT-2001-00271, RAAG.

(i) NðXÞ, the smallest integer nd1 such that any divisor of degree n is linearly equivalent to a totally real e¤ective divisor (by [11] Theorem 2.7, we know that NðXÞis finite),

(ii) MðXÞ, the smallest integermd1 such that any divisorDof degree 2msuch that the degree ofDon each connected component ofXðRÞis even, is linearly equiva- lent to a totally non-real e¤ective divisor. If such an integer does not exist, then MðXÞ ¼ þy.

The principal goal of the paper is to bound the previous invariants in terms ofgand s. The problem forNðXÞwas raised by Scheiderer in [11].

We briefly describe the structure of the paper. In Section 2, we show that gcMðXÞc2g:

Moreover, ifXis a real rational curve or a real elliptic curve, thenMðXÞ ¼1. Using this, we also prove that ifXJP_Rⁿ,nd2, is a non-degenerate linearly normal curve of degree d with no pseudo-line in XðRÞ (see the Section 2 for the corresponding definitions), and ifX satisfies one of the two following conditions

(i) X is rational or elliptic, (ii) gd2 anddd4g,

then XðRÞ is a‰ne in P_Rⁿ, i.e. there exists a real hyperplaneH such thatHðRÞV XðRÞ ¼q.

In Section 3, we extend a result proved in [6] forM-curves, toðM1Þ-curve:

NðXÞc2g1:

Under the assumption of a conjecture of Huisman [9] on unramified curves, we fur- ther extend this result toðM2Þ-curves, the bound being slightly di¤erent.

In the last section of the paper, we give a large family of curves for which the invari- antNis explicitely calculated. For these computations, we use the results established in Sections 4 and 5.

In Section 4, we prove a stronger version of the Cli¤ord inequality for real hyper- elliptic curves, which sharpen Huisman’s general result for real curves [8]: if X is a real hyperelliptic curve such that s02 and DADivðXÞ is an e¤ective and special divisor of degreed, then

dimjDjc1

2ðddðDÞÞ;

withdðDÞthe number of connected componentsC ofXðRÞsuch that the degree of the restriction ofDtoCis odd.

Section 5 deals with the existence of complete linear systems of degreedand dimen- sionrd1 onX.

The author wishes to express his thanks to D. Naie and J. van Hamel for several helpful comments concerning the paper. I also thank J. Huisman for bringing my attention to his work on real algebraic curves.

1 Preliminaries

We recall here some classical concepts and more notations that we will be using throughout this paper.

Let X be a real curve. We will denote by XC the base extension ofX to C. The group DivðXCÞof divisors onXCis the free abelian group on the closed points ofXC. The Galois group GalðC=RÞacts on the complex varietyXC and also on DivðXCÞ.

We will always indicate this action by a bar. IfPis a non-real point ofX, identifying DivðXÞand DivðXCÞ^GalðC=RÞ, thenP¼QþQwithQa closed point ofXC.

IfDis a divisor onX orX_C, we will denote by½Dits class in the Picard group, and byOðDÞits associated invertible sheaf. The dimension of the space of global sections of this sheaf will be denoted bylðDÞforDonX, and bylCðDÞforDonXC.

We will always denote by C₁;. . .;C_s the connected components of XðRÞ. Let DADivðXÞ, and denote by deg_CiðDÞ the degree of the restriction ofDto C_i. Fol- lowing [4], we will denote bycthe surjective morphism

PicðXÞ ! ðZ=2Þ^s;

½D 7! ð. . .;deg_CiðDÞmod 2;. . .Þ;

and we will write dðDÞfor the number of connected components C of XðRÞ such that deg_CðDÞis odd. The connected components of Pic^dðXÞ, the subgroup of divi- sor classes of PicðXÞ of degree d, correspond to the fibres of the restriction ofcto Pic^dðXÞ. Letu¼ ðu1;. . .;u_sÞAðZ=2Þ^s, we will denote by Uðd;u₁;. . .;u_sÞ ¼Uðd;uÞ the connected component of Pic^dðXÞ that corresponds to c¹ðuÞ. Obviously, Uðd;u₁;. . .;u_sÞ0 q if and only if Ps

i¼1u_i1d mod 2. We will also denote the co- ordinates ofu¼ ðu1;. . .;u_sÞAðZ=2Þ^sbyc_iðuÞ ¼u_i.

Let J be the Jacobian ofX. It is well known that Pic⁰ðXÞcan be identified with JðRÞsince XðRÞ0 q. We will denote by JðRÞ₀ the connected component of the identity ofJðRÞ. ThenJðRÞ₀¼Uð0;0;. . .;0Þ([11] Lemma 2.6).

We now reformulate the definition of the invariantsN andM.

Definition 1.1. (i)NðXÞis the smallest integernd1 such that for any real pointP, and for anyaAJðRÞ, there existP1;. . .;PnAXðRÞ, such thata¼Pn

i¼1½PiP, and (ii)MðXÞis the smallest integermd1, such that for any real closed pointP, and for anyaAJðRÞ₀, there exist non-real pointsQ1;. . .;Qmsuch thata¼Pm

i¼1½Qi2P.

If such an integer does not exist, thenMðXÞ ¼ þy.

2 Divisors with a complex support

In this section, we bound the invariantMðXÞfrom above and from below, and give a geometric consequence.

The following proposition justifies the definition of the invariantM.

Proposition 2.1.Let P be a real point of X andaAJðRÞ₀.There is an integer md1 and non-real points Q₁;. . .;Q_msuch thata¼Pm

i¼1½Qi2P.

Proof. Let Pbe a real closed point of X andaAJðRÞ₀. Since JðRÞ₀ is a divisible group, there isbAJðRÞ₀such that 2b¼a. By Riemann–Roch, the map

j_d:ðS^dXÞðRÞ !Pic^dðXÞ

is surjective for ddg, whereS^dX denotes the symmetric d-fold product ofX over R. Hence there existsDan e¤ective divisor of degreegsuch thatbþ ½gP ¼ ½D. By Riemann–Roch, there is an integer k such that the divisor kP is very ample as a complex divisor, and also as a real divisor, sincekPADivðXÞ. HenceDþkPis also very ample.

Let cdenote the embedding of X inP^k

Rassociated to the linear system jDþkPj.

Let S be the quadric hypersurface of P_R^k with equation x²₀þ þx_k²¼0. Thus 2Dþ2kPis linearly equivalent to the e¤ective divisorD⁰of degree 2ðgþkÞobtained by intersecting S andX. Since SðRÞ ¼q,D⁰ is totally non-real. Hencea¼ ½D⁰

½2ðgþkÞP, and the result follows. r

The method of the previous proof allows us to give an upper bound forMðXÞin terms ofg. The following theorem gives a better result.

Theorem 2.2.Let X be a curve of positive genus.We have MðXÞc2g.

Proof. LetPbe a real point ofX andV¼XðCÞnXðRÞ, whereXðCÞdenote the set of closed points of XC.XðRÞis seen as a subset of XðCÞ. By Riemann–Roch, the mapXðCÞ^g!Pic^gðXCÞis surjective. Moreover, the mapS^gX!Jis well known to be a birational morphism of complete varieties. The imageUof the map

V^g!JðCÞ;ðQ1;. . .;Q_gÞ 7!X^g

i¼1

½QiP;

contains therefore an open dense subset ofJðCÞ. ThusUþU¼JðCÞ. The image of the norm mapN:JðCÞ !JðRÞ;a7!aþa, isJðRÞ₀(see [11]). SoNðUÞ þNðUÞ ¼

JðRÞ₀, andMðXÞc2g. r

Since any two divisors with the same degree on a rational real curve are linearly equivalent, we trivially get the following proposition:

Proposition 2.3.Let X be a real rational curve,then MðXÞ ¼1.

For real elliptic curves, the result of Theorem 2.2 can be improved.

Theorem 2.4.Let X be a real elliptic curve,then MðXÞ ¼1.

Proof. LetPbe a real point ofX andaAJðRÞ₀. Arguing as in the proof of Propo- sition 2.1, there isbAJðRÞ₀such that 2b¼aandbþ ½P ¼ ½P0, withP₀a real point.

Then

a¼ ½2P0 ½2P:

The linear system j3P0j gives a closed immersion XJP_R². Using Riemann–Roch and after linear changes of coordinates, we obtain a closed immersion j:X !P²

R

such that the image is the curve

y² ¼ ðxaÞRðxÞ;

withaARandRðxÞAR½xa monic and separable polynomial of degree 2. The point P₀goes to the point at infinityð0:1:0Þon the y-axis (see [5] Proposition 4.6, p. 319).

If we project fromP₀onto thex-axis, we get a finite morphism f :X!P_R¹ of degree 2, sendingP₀toy, and being ramified at least at one more real point ofP_R¹, besides y. In fact, f may be defined using the linear systemj2P0j. Since f is ramified with order 2 aty, then locally on one side ofythe fiber overP_R¹ðRÞis totally real and on the other side the fiber is totally non-real. In particular, there exists lAP_R¹ðRÞ such that f¹ðlÞ ¼ fQg, with Q a non-real point of X. Then½2P0 ¼ ½Q anda¼

½Q ½2P. r

For a given complete linear system of degree su‰ciently big, an upper bound exists for the least degree of the real part of divisors in the linear system.

Corollary 2.5. For any complete linear systemjDjwithdegðDÞd4gþdðDÞif gd2, degðDÞd2þdðDÞif gAf0;1g,there exists D⁰AjDjsuch that the real part of D⁰has degreedðDÞ.

Proof.We give the proof only for the casegd2. LetP₁;. . .;P_dðDÞbe some real points belonging to the connected components ofXðRÞwhere the degree ofDis odd, and such that no two of them belong to the same connected component ofXðRÞ. We set d ¼degðDÞ. We remark thatddis necessarily even. By Theorem 2.2,DPdðDÞ

i¼1 P_i is linearly equivalent to a totally non-real e¤ective divisor and the proof is done. r

We give a lower bound for the invariantMðXÞ.

Proposition 2.6.Assume gd2.Then MðXÞdg.

Proof. LetPAXðRÞand consider the divisorD⁰¼KP, whereK denotes the ca- nonical divisor. ChooseP⁰0PAXðRÞbelonging to the same connected component of XðRÞas P. SinceX is not rational, using the fact thatlðPP⁰Þ ¼0, it follows that

lðD⁰þP⁰Þ ¼g1¼lðD⁰Þ:

Hence P⁰ is a base point of the linear system jD⁰þP⁰j. Since D⁰þP⁰ has degree 2g2 and has an even degree on each connected component ofXðRÞ(see [9] Prop-

osition 2.1), we easily see thatMðXÞ>g1. r

We give now a geometric consequence of the previous results. Let XJP_Rⁿ be a non-degenerate real curve, i.e.Xis not contained in a real hyperplane ofP_Rⁿ. We will say that XðRÞis a‰ne inP_Rⁿ if there exists a real hyperplaneH such thatHðRÞV XðRÞ ¼q. In this case XðRÞis a real algebraic subvariety ofAⁿ

RðRÞ ¼Rⁿ in the sense of [2]. Since the real hypersurfaceSofPⁿ

Rwith equationx₀²þ þx²_n¼0 has no real points,XðRÞis always contained in an a‰ne open subset ofPⁿ

R. More pre- cisely the image of XðRÞ by the 2-uple embedding is a‰ne inP

1=2ðnþ1Þðnþ2Þ1

R . We

may wonder if XðRÞis already a‰ne inP_Rⁿ. Recall thatX is linearly normal if the restriction map

H⁰ðP_Rⁿ;Oð1ÞÞ !H⁰ðX;Oð1ÞÞ

is surjective. LetCbe a connected component ofXðRÞ. The componentCis called a pseudo-line if the canonical class ofCis nontrivial inH₁ðP_RⁿðRÞ;Z=2Þ. Equivalently, C is a pseudo-line if and only if for each real hyperplane H, HðRÞintersects C in an odd number of points, when counted with multiplicities (see [9]). So a necessary condition forXðRÞto be a‰ne inP_Rⁿ is thatXðRÞhas no pseudo-line.

Proposition 2.7. Let XJPⁿ

R, nd2, be a non-degenerate linearly normal curve of degree d such that XðRÞ has no pseudo-line. If X satisfies one of the two following conditions

(i) X is rational or elliptic, (ii) gd2and dd4g, then XðRÞis a‰ne inP_Rⁿ.

Proof.A hyperplane section has even degree on each connected component ofXðRÞ and its degreed2MðXÞ. The results follows from Corollary 2.5 and the linear nor-

mality. r

Example 2.8. LetX be an elliptic quartic curve inP³_R with only one real connected component. ThenXðRÞis a‰ne inP_R³ sinceX satisfies the hypotheses of the prop- osition:X is a complete intersection andd is even (use Bezout’s theorem).

Proposition 2.9.Let XJPⁿ

Rbe a non-degenerate curve of degree dc2n1such that XðRÞ has no pseudo-line and g¼dn. If ncdcnþ1 or dc⁴₃n then XðRÞ is a‰ne inPⁿ

R.

Proof. Let H be a hyperplane section of X. By Cli¤ord’s inequality and since dc2n1, H is non-special (see Section 4). By Riemann–Roch, g¼ddimjHj.

Consequently dimjHj ¼n and X is linearly normal. The proof follows now from

Proposition 2.7. r

Example 2.10. Let X be a smooth quartic curve in P_R². Then X is the canonical

model of a curve of genus 3. By [4], X has always odd theta-characteristics that are in one-to-one correspondence with the real bitangent lines toX. Since the degree of X is 4, a real bitangent line toX intersects the curve XC only at the two points of tangency. If these two points are non-real and switched by the complex conjugation, thenXðRÞis a‰ne inP²

R. If the points are real, we may move the line to get a line which does not intersectXðRÞ, XðRÞis again a‰ne inP²

R. Notice that the conclu- sion cannot be deduced from Proposition 2.7.

3 Divisors with real support

This section is dedicated to the study of the invariantNðXÞ. We clearly have Proposition 3.1.If X is a real rational curve or a real elliptic curve,then

NðXÞ ¼1:

Hence, in the remainder of this section we will assume thatg>1, and use the invari- antedefined by:

e¼

1

2ðgsÞ if gseven;

1

2ðgsþ1Þ if gsodd:

(

Let us state the principal result of this section:

Theorem 3.2.Any complete linear system of degreeds1þg contains a divisor whose non-real part has degreec2e.

Proof. Let Dbe a divisor of degree dds1þg. We will prove that Dis linearly equivalent to an e¤ective divisor, whose non-real part has degreec2e.

Let P be a real point and a¼ ½DdPAJðRÞ. We fix R₁;. . .;Rg2e in g2e distinct components amongC₁;. . .;C_s. To simplify the proof, we setR_iAC_i. Let us denoteb¼aþPg2e

i¼1 ½PR_i. Consider the restriction to Pic⁰ðXÞof the morphismc defined in Section 1, then it clearly induces an isomorphismJðRÞ=JðRÞ₀FðZ=2Þ^s1. Hence there existPg2eþ1;. . .;Pg2eþs1AXðRÞsuch that

b¼X^s1

j¼1

½Pg2eþjP þb₀;

withb₀AJðRÞ₀.

By Riemann–Roch, the natural map ðS^gXÞðRÞ !Pic^gðXÞ is surjective, S^dX denoting the symmetric d-fold product of X over R. Moreover if ½D⁰ ¼ ½D⁰⁰ in Pic^dðXÞ, then deg_CiðD⁰Þ1deg_CiðD⁰⁰Þmod 2 fori¼1;. . .;s. LetuAðZ=2Þ^ssuch that c_iðuÞ ¼1 for i¼1;. . .;g2e and cg2eþ1ðuÞ ¼0. Consequently, if ½D⁰AUðg;uÞ, thenD⁰is linearly equivalent to the e¤ective divisor

X

g2e

i¼1

PiþX^e

i¼1

Qi; where

1) P_iAC_i, 1cicg2eand,

2) Q_i is either a non-real point or a sum of two real points contained in the same connected component ofXðRÞ,i¼1;. . .;e.

The translation by ½Pg2e

i¼1 Ri 2e½Pis a bijection between Uðg;uÞandJðRÞ₀¼ Uð0;0;. . .;0Þ, hence

b₀þ X^g2e

i¼1

Ri

" #

þ2e½P ¼X^g2e

i¼1

½Pi þX^e

i¼1

½Qi:

Finally,

a¼^s1þg2eX

i¼1

½PiP þX^e

i¼1

½Qi2P

and the proof is done. r

The above theorem allows to give an upper bound forM-curves orðM1Þ-curves.

Corollary 3.3.Let X be an M-curve or anðM1Þ-curve.Then NðXÞcs1þg:

In [6], it is shown that NðXÞc2g1 for M-curves. Following the method used in [6], we will now show that the result of Theorem 3.2 may be improved in the case s1gþ1 mod 2.

Letsd2. By Theorem 3.2, we already know that for every complete linear system jDjof degreeds1þg, there existsD⁰AjDjsuch that the non-real part ofD⁰has degreec2e. We would like to extend the result to linear systems of degree gþd, 0cdcs2, under certain conditions on the invariantd.

Proposition 3.4. Assume degðDÞ ¼gþd for dAf0;. . .;s2g. If dðDÞdsd

1

2ð1 ð1Þ^sgÞ, then there exists D⁰AjDj such that the non-real part of D⁰ has degreec2e.

Proof.The proof depends on the parity ofsg.

First, assume thatsg is odd. Fori¼1;. . .;s, letu_iAðZ=2Þ^s such thatc_jðuiÞ ¼ 1d_i;j (dis Kronecker’s symbol). By Riemann–Roch, any divisor inUðg;u_iÞis lin-

early equivalent to an e¤ective divisor whose non-real part has degreec2e. We translate D by D⁰⁰, with D⁰⁰ a totally real e¤ective divisor of degree d such that

½DD⁰⁰AUðg;uiÞ for ai. We havedðDÞ ¼gþd mod 2. Hence there exists kAZ such that dðDÞ þ2k¼gþd. Moreover gþd¼sd1 mod 2, hence gþd ¼ sd1þ2r, withrAZ. By a closer look at these identities, we see thatkandrare non-negative. Consequently

dðDÞ ¼2ðrkÞ þsd1: ð1Þ

By the hypothesisdðDÞdsd1. Hencel¼rkd0 and by (1),

ðsdðDÞ 1Þ þ2l¼d: ð2Þ

We remark thatsdðDÞcorresponds to the number of connected componentsCof XðRÞwhere deg_CðDÞis even. Ifs0dðDÞ, then we choose a componentC_isuch that deg_CiðDÞis even, and by (2), we take asD⁰⁰ a divisor that cuts out schematically a point on the components Cj0Ci where deg_CjðDÞis even, and a point with multi- plicity 2l on Ci. Then ½DD⁰⁰AUðg;uiÞ. If s¼dðDÞ, thend ¼2l1 is odd, and we takeD⁰⁰¼dP1, withP1AC1. Again½DD⁰⁰AUðg;u1Þ.

Second, assume thatsgis even.

The situation is simpler since we know that any divisor in Uðg;uÞ, with u¼

ð1;. . .;1ÞAðZ=2Þ^s, is linearly equivalent to an e¤ective divisor whose non-real part

has degreec2e. So we translateDbyD⁰⁰withD⁰⁰a totally real e¤ective divisor of degreed, such that½DD⁰⁰AUðg;uÞ. By the same arguments as before,

dðDÞ ¼2ðrkÞ þsd; ð3Þ

for some non-negative integers r and k. If we assume that dðDÞdsd, then l¼ rkd0, and by (3),

ðsdðDÞÞ þ2l¼d: ð4Þ

Again sdðDÞ corresponds to the number of connected components C of XðRÞ where deg_CðDÞis even. ForD⁰⁰, we take the sum of any real point with multiplicity 2l, with a divisor whose support consists of a unique point in each of the component

CofXðRÞ, where deg_CðDÞis even. r

Corollary 3.5. Assume sg is odd and sd2. Any complete linear system of degree ds2þg contains a divisor whose non-real part has degreec2e.

Proof.Using the previous proposition, we only have to prove that ifDis a divisor of degree gþs2, then dðDÞd1. If dðDÞ ¼0, then gþs2 must be even, contra-

dicting the hypotheses. r

Let us state a nice consequence of the previous results:

Theorem 3.6.Let X be an M-curve or anðM1Þ-curve.Then NðXÞc2g1.

Equivalently, the theorem says that, for anM-curve or anðM1Þ-curve, the natural mapXðRÞ^2g1!Pic^2g1ðXÞis surjective.

(MC2)-curves and unramified real curves in odd-dimensional projective spaces.LetX be real curve and DADivðXÞ. For D¼P

n_iP_iP

m_jQ_j, with n_i andm_j positive, and the sum taken over distinct closed points of X, we define D_red¼P

P_iP Q_j. We also define the weight of Dto be the natural numberwðDÞ ¼degðDD_redÞ. If XJP_Rⁿ,nd1, is non-degenerate, we say thatXis unramified if for each hyperplane H ofPⁿ

R, we havewðHXÞcn1.

The corresponding notion of an unramified complex algebraic curve in complex projective space is well understood. Indeed, any unramified complex algebraic curve is a rational normal curve and conversely [3]. Over R, the situation is di¤erent and Huisman has given the following conjecture (see [9] Conjecture 3.6):

Conjecture. Let nd3be an odd integer and XJP_Rⁿ be a non-degenerate real alge- braic curve of positive genus.If X is unramified,then X is an M-curve.

We relate this conjecture and the invariantNstudied in this paper.

Theorem 3.7.Let X be anðM2Þ-curve.Assuming the above conjecture,we get:

(i) NðXÞc3g1,if g is even,and (ii) NðXÞc3g,if g is odd.

Proof.LetPAXðRÞandaAJðRÞ. Recall thats¼g1 and thatC1;. . .;Cg1denote the connected components ofXðRÞ. We may assume thatPAC1.

Assumegis even. LetD¼P2þ þPg1þQbe an e¤ective divisor withPiACi

for i¼2;. . .;g1, and Q be a non-real point. In fact ½DAUðg;0;1;. . .;1Þ. Let

D⁰¼Dþ ðgþ1ÞP. ThenD⁰ is very ample and the linear system jD⁰j allows us to embed X in P

gþ1

R . Using the above conjecture, X is not unramified. Consequently there is an hyperplane H ofP

gþ1

R such that HX¼Pr

i¼1niRiþPt

j¼1mjQj, where the sum is taken over distinct points. TheRi are real points and theQj are non-real points. Moreover,

X^r

i¼1

niþ2X^t

j¼1

mj¼2gþ1 ð5Þ

and

wðHXÞ ¼X^r

i¼1

ðni1Þ þ2X^t

j¼1

ðmj1Þdgþ1: ð6Þ

Since deg_CiðD⁰Þ is odd for i¼1;. . .;s, each connected component of XðRÞ is a pseudo-line. It follows that

wðHXÞ ¼degððHXÞ ðHXÞ_redÞcð2gþ1Þ ðg1Þ ¼gþ2: ð7Þ Using (5), (6) and (7), we getgþ2dðrþ2tÞ þ2gþ1dgþ1 and

gdðrþ2tÞdg1:

Sincerdg1, we havet¼0 andD⁰ is linearly equivalent to a totally real e¤ective divisor. By Theorem 3.2, there are 2g4 pointsP1;. . .;P2g4AXðRÞandQa non- real point or a sum of two real points contained in the same connected component of XðRÞ, such that

a¼ X^2g4

i¼1

½PiP

þ ð½Q ½2PÞ:

Moreover, looking at the proof of Theorem 3.2, we may choose PiACi for

i¼2;. . .;g1. Writing a¼aþ ðgþ1Þ½P ðgþ1Þ½P, the statement follows from

the above construction.

If we assume that g is odd, the proof is similar but for D⁰¼Dþ ðgþ2ÞP

here. r

Assuming the above conjecture, we may also obtain a more general result.

Proposition 3.8. Let X be a real curve such that scg1.Any complete linear sys- tem of degree dsþ2gþ¹₂ð1 ð1Þ^gÞ contains a divisor whose non-real part has degreec2e2.

The above method allows us to have a description of a linear system with one non- real point less. Unfortunately this method is rigid; a repetition of this method does not give a description of a linear system with only real points.

4 Cli¤ord’s inequality and linear systems on real hyperelliptic curves In this section, we study the family of special linear systems on real algebraic curves.

Let DADivðXÞ and K be the canonical divisor. If lðKDÞ>0,D is said to be special. If not, Dis said to be non-special. By Riemann–Roch, if degðDÞ>2g2 thenDis non-special. The classical Cli¤ord inequality states that the dimension of a nonempty special complete linear system on a curve is bounded by half of its degree (see [5] Theorem 5.4, p. 343). We now recall the Cli¤ord inequality for real curves given by Huisman ([8] Theorem 3.1).

Theorem 4.1.Let DADivðXÞbe an e¤ective divisor of degree d.The following state- ments hold.

(i) If dþdðDÞ<2s,thendimjDjc¹₂ðddðDÞÞ.

(ii) If dþdðDÞd2s,thendimjDjcdsþ1.

A real hyperelliptic curve is a real curve X such that XC is hyperelliptic, i.e. XC

has a g₂¹ (a linear system of dimension 1 and degree 2). As always, we assume that XðRÞ0 qand moreover thatgd2.

Lemma 4.2.Let X be a real hyperelliptic curve.Then X has a unique g¹₂.

Proof. By [5] Proposition 5.3, X_C has a unique g₂¹. LetD be an e¤ective divisor of degree 2 on XC satisfying jDj ¼g¹₂. Since this unique g¹₂ is also complete and X is defined over R, we have jDj ¼g₂¹. Let PAXðRÞ, we also denote by P the corre- sponding closed point of XC. Since lCðDPÞ>0, we may assume thatD¼PþQ with Qa closed point of XC. Then ½PþQ ¼ ½PþQ in PicðXCÞandQ¼Q, since XCis not rational. HenceD¼Dand sincelðDÞ ¼lCðDÞ, the proof is done. r This g₂¹ induces an involution, denoted by {, on the closed points of X. A real hyperelliptic curveX is said to be respected by the involution (we will abbreviate by r.b.i.), if for any real pointP,Pand{ðPÞbelong to the same real connected compo- nent. Most real hyperelliptic curves are r.b.i.

Proposition 4.3.Let X be a real hyperelliptic curve such that X is not r:b:i.Then X is given by the real polynomial equation y²¼ fðxÞ, where f is a monic polynomial of degree2gþ2,with g odd,and where f has no real roots.In particular,the number of connected components of XðRÞis2.

Proof. Using the g₂¹, we easily see that an a‰ne model of X is given by the real equation y²¼ fðxÞ, with degðfÞ ¼2gþ2. SinceXis not r.b.i., f cannot have a real root. We may assume that f is monic sinceXðRÞ0 q. Ifg is even, thens¼1 ([4]

Proposition 6.3), contradicting the hypotheses. If g is odd, then XðRÞ has 2 con-

nected components exchanged by{. r

We give now the Cli¤ord inequality for real hyperelliptic curves which are r.b.i.

Theorem 4.4.Let X be a real hyperelliptic curve that is r:b:i.and let DADivðXÞbe an e¤ective and special divisor of degree d.Then

dimjDjc1

2ðddðDÞÞ:

Proof. The classical Cli¤ord inequality allows us to assume thatdðDÞd1. We may further assume thatC₁;. . .;C_dðDÞare the connected components ofXðRÞ, where the degree ofDis odd. LetD⁰cDbe the greatest e¤ective common subdivisor ofDand {ðDÞ with the property thatjD_r⁰j ¼^degðD₂ ^r0Þg₂¹,D_r⁰ denoting the real part of D⁰. Write

D⁰⁰¼DD⁰. SinceXis r.b.i., thendðD⁰Þ ¼0. So, there are real pointsP₁;. . .;P_dðDÞ such thatP1þ þP_dðDÞcD⁰⁰andPiACi,i¼1;. . .;dðDÞ. We remark that a) dþdðDÞc2g2 sinceDis special ([9] Theorem 2.3),

b) {ðPiÞBSuppðD⁰⁰ÞorPiis a fixed point for{such that 2Piis not a subdivisor ofD⁰⁰, i¼1;. . .;dðDÞ.

Letobe a global di¤erential form onX, such that divðoÞdD. Then divðoÞdDþ {ðP1Þ þ þ{ðPdðDÞÞ, sinceK¼ ðg1Þg¹₂anddþdðDÞc2g2. HencelðKDÞ ¼ lðK ðDþ{ðP1Þ þ þ{ðPdðDÞÞÞÞ, andDþ{ðP1Þ þ þ{ðPdðDÞÞis also special. By Riemann–Roch,

dimjDj dimjK ðDþ{ðP1Þ þ þ{ðPdðDÞÞÞj ¼dgþ1; ð8Þ and

dimjDþ{ðP1Þ þ þ{ðPdðDÞÞj dimjK ðDþ{ðP1Þ þ þ{ðPdðDÞÞÞj

¼dþdðDÞ gþ1: ð9Þ

Since Dþ{ðP1Þ þ þ{ðPdðDÞÞ is e¤ective and special, by the classical Cli¤ord inequality, we get

dimjDþ{ðP1Þ þ þ{ðPdðDÞÞjc1

2ðdþdðDÞÞ:

Replacing in (9), we have

dimjK ðDþ{ðP1Þ þ þ{ðPdðDÞÞÞjc1

2ðdþdðDÞÞ ðdþdðDÞÞ þg1

¼g11

2ðdþdðDÞÞ: ð10Þ

Finally, combining (8) and (10), we get dimjDjc1

2ðddðDÞÞ: r

Theorem 4.5.Let X be a real hyperelliptic curves that is not r:b:i.and let DADivðXÞ be an e¤ective and special divisor of degree d.Then

dimjDjc1

2ðddðDÞÞ;

except whenjDj ¼rg¹₂,with0<r<g1and r odd,in which casedimjDj ¼r¼¹₂d.

Proof.We recall that under these hypothesess¼2. IfdðDÞ ¼0, the classical Cli¤ord inequality applies.

So let us assumedðDÞ ¼1. As in the previous proof, we writeD¼D⁰þD⁰⁰, with D⁰ the greatest e¤ective common subdivisor ofDand{ðDÞ, and D⁰⁰ e¤ective. Since dðDÞ ¼1 and since the two connected components ofXðRÞare exchanged by{, we easily see that there exists a real pointPin the support ofD⁰⁰. Repeating the proof of the previous theorem, we get the result.

Now, if dðDÞ ¼2, then the above arguments give the proof, except when D is invariant by{. In this casejDj ¼rg₂¹,ris odd and dimjDj ¼r. r We give some applications of the previous theorems. We know that Castelnuevo’s inequality is one of the consequences of the (complex) Cli¤ord inequality (see [3] cor- ollary p. 251). Hence, we obtain a Castelnuevo inequality for real hyperelliptic curves.

Proposition 4.6.Let nd2be an integer and XJP_Rⁿ be a non-degenerate real hyper- elliptic curve r:b:i.Let d be the degree of X anddbe the number of pseudo-lines of X.

Assume d<2nþd.Then

gcdn;

with equality holding if and only if X is linearly normal.

Proof. LetH be a hyperplane section ofX. Then dimjHjdn>¹₂ðddðHÞÞby the hypotheses. Theorem 4.4 says thatH is non-special and by Riemann–Roch,

g¼ddimjHjcdn:

Clearly, the previous inequality becomes an equality if and only if the map H⁰ðP_Rⁿ;Oð1ÞÞ,!H⁰ðX;Oð1ÞÞis an isomorphism. r Proposition 4.7. Let X be a real hyperelliptic curve r:b:i. Let D¼P1þ þPr, 0crcg, such that P1;. . .;PrAXðRÞ and such that no two of them belong to the same connected component of XðRÞ.ThenlðDÞ ¼1.

Remark 4.8.In the previous proposition, ifXis any real algebraic curve andr¼s, by Theorem 4.1, we can only say thatlðDÞc2.

Let us set some more notations. Fordd0, letS^dX denote the symmetric d-fold product of X over R. We have a natural map j_d:ðS^dXÞðRÞ !Pic^dðXÞ. Write W_dðRÞ ¼Imðj_dÞ for the real part of the subvariety W_d of Pic^dðXCÞ(see [1]), and yðRÞJJðRÞfor the real part of the theta divisor.

Proposition 4.9. Let X be a real hyperelliptic curve r:b:i. such that sdg1. If uAðZ=2Þ^s satisfies Ps

i¼1c_iðuÞdg1,then Wg1ðRÞVUðg1;uÞdoes not contain any singularity of Wg1.

Proof. If Ps

i¼1c_iðuÞdg, then Wg1ðRÞVUðg1;uÞ ¼q. Let ½DAWg1ðRÞV Uðg1;uÞ, where uAðZ=2Þ^s satisfies Ps

i¼1ciðuÞ ¼g1. Then lðDÞ>0 and we may assume that Dis e¤ective. By [1] Corollary 4.5, p. 190, the singular points of Wg1 correspond to the complete linear systems of dimensiond1 and degreeg1.

By Theorem 4.4, dimjDj ¼0, hence the result. r

We may extend the previous proposition in any degree.

Proposition 4.10.Let X be a real hyperelliptic curve r:b:i:,and let d be a non-negative integercs.If uAðZ=2Þ^ssatisfiesPs

i¼1c_iðuÞdd,then W_dðRÞVUðd;uÞdoes not con- tain any singularity of W_d.

We prove a result similar to Theorem 3.1 in [7].

Proposition 4.11.Let X be a real hyperelliptic curve r:b:i:which is anðM2Þ-curve.

Then C1 Cg1is homeomorphic to the real part of the theta divisor contained in the neutral component JðRÞ₀ of the real part of the Jacobian.

Proof. By Proposition 4.9, C1 Cg1 is homeomorphic to the part of Wg1ðRÞ contained in Uðg1;1;. . .;1Þ. We may easily find a theta-characteristic kAPic^g1ðXÞ(i.e. 2k¼ ½K) such thatkAUðg1;1;. . .;1Þ. By Riemann’s theorem (see [1] p. 27),Wg1¼yþkand the proof is straightforward. r Remark 4.12. Most of the results of this section are also valid for any real algebraic curve withsdg(see [8] and Theorem 4.1).

The following result states a remarkable property of some special linear systems.

Proposition 4.13.Let X be a real hyperelliptic curve r:b:i.Let DADivðXÞ be a spe- cial e¤ective divisor of degree d satisfyingdimjDj ¼¹₂ðddðDÞÞ.ThenjDjcontains a totally real divisor.

Proof. Firstly, we assume thatdcg. A consequence of the geometric version of the Riemann–Roch theorem is that any completeg_d^r onXCis of the form

rg¹₂þP₁þ þPd2r;

where no two of thePi are conjugate under{. Hence the complete linear systemjDj on XC is of this form, with r¼¹₂ðddðDÞÞ. Since D¼D, we haveD⁰¼P1þ þ P_d2rADivðXÞ. It follows thatjDjis of the form

1

2ðddðDÞÞg₂¹þD⁰;

whereD⁰is an e¤ective divisor of degreedðDÞ. Any divisor ing¹₂is linearly equivalent

toPþ{ðPÞwherePAXðRÞ. SinceX is r.b.i., we havedðD⁰Þ ¼dðDÞandD⁰is totally real.

Secondly, let d >g. The residual divisor KD is of degree 2g2d<g2.

Since the degree ofK is even on each connected component ofXðRÞ,KDis also special and satisfiesdðKDÞ ¼dðDÞ. So, dimjKDj ¼¹₂ðddðDÞÞ dþg1¼

1

2ð2g2ddðDÞÞ ¼¹₂ðdegðKDÞ dðKDÞÞ. We may apply the first part of the proof toKDto obtain that

½KD ¼ 1

2ð2g2ddðDÞÞðPþ{ðPÞÞ

þ ½P1þ þP_dðDÞ; ð11Þ where P;P₁;. . .;P_dðDÞAXðRÞ and no two of the P_i are conjugate under {. Since jKj ¼ ðg1Þg₂¹, then½K ¼ ½ðg1ÞðPþ{ðPÞÞ. From (11), we get

½D ¼ 1

2ðdþdðDÞÞðPþ{ðPÞÞ

½P1þ þP_dðDÞ:

Then½D ½{ðP1Þ þ þ{ðPdðDÞÞ ¼ ½¹₂ðddðDÞÞðPþ{ðPÞÞand the proof is done.

r

5 Existence of special linear systems of dimensionron real curves Curves are classified by their genus. But we may further subdivide them according to whether or not they possess completeg_d^r, i.e. complete linear systems of degreed and dimension rd1, for various d andr. For complex curves, we may find numerous results on this subject, it is a part of the Brill–Noether theory. This section deals with these problems for real curves.

5.1 Complete linear systems of dimensionron real curves.

Definition 5.1.ForX a real curve andra positive integer, we set:

(i) r_CðX;rÞ ¼inffd ANjXChas a completeg_d^rg.

(ii) r_RðX;rÞ ¼inffd ANjX has a completeg_d^rg.

Forgd0 andr>0, we set:

(iii) r_Cðg;rÞ ¼supfr_CðX;rÞ jX is a curve of genusgg.

(iv) r_Rðg;rÞ ¼supfr_RðX;rÞ jX is a curve of genusgg.

Remark 5.2.It is easy to check that theg_r^r

RðX;rÞ andg_r^r

CðX;rÞ are necessarily complete.

Using the Riemann–Roch formula, it is easy to show thatr_Rð0;rÞ ¼r_Cð0;rÞ ¼r, and that r_Rð1;rÞ ¼r_Cð1;rÞ ¼rþ1. In the remainder of the section we will assume thatgd2.

Remarks 5.3.If r>g1, by the classical Cli¤ord inequality, a completeg_d^r is non- special and r_RðX;rÞ ¼gþr. From now on, we will also assume that rcg1 in order to deal with special linear systems. By the classical Cli¤ord inequality, and since there exist non-special linear systems of degreegþr, we have

2rcr_RðX;rÞcgþr:

Since the canonical divisor is invariant by the complex conjugation, by the previous inequality, we have the equalities

r_RðX;g1Þ ¼r_Rðg;g1Þ ¼r_CðX;g1Þ ¼r_Cðg;g1Þ ¼2g2:

From the classical theory of special linear systems (see [1] Theorem 1.1 p. 206, The- orem 1.5 p. 214), we may see r_Cðg;rÞas the smallest integerd such that the Brill–

Noether numberrðg;r;dÞ ¼g ðrþ1ÞðgdþrÞis non-negative.

Now, we state the principal result of this section.

Theorem 5.4. Let X be a real curve and let r be an integer such that 1crcg1.

Then

(i) r_RðX;rÞcr_Rðg;rÞcgþr1,and (ii) r_CðX;rÞcr_RðX;rÞc2r_CðX;rÞ 2r.

Proof.For (i), letDADivðXÞbe an e¤ective divisor of degreeg1r. ChoosingD general, we have lðDÞ ¼1. Then the residual divisor KDADivðXÞand satisfies lðKDÞ ¼1 ðg1rÞ þg1¼rþ1, and we get the first assertion.

As for (ii), clearlyr_CðX;rÞcr_RðX;rÞ, since for any divisorDADivðXÞwe have lðDÞ ¼lCðDÞ. It remains to show that

r_RðX;rÞc2r_CðX;rÞ 2r:

Let d ¼r_CðX;rÞ and let DADivðXCÞ be an e¤ective divisor of degree d such that dimCjDj ¼r. Let P1;. . .;Pr be real points of X. We also denote by P1;. . .;Pr

the corresponding closed points of XC. We may choose P1;. . .;Pr such that lðP1þ þPrÞ ¼1 andlCðDP1 PrÞ ¼1. Moreover, we may assume that D¼D⁰⁰þD⁰, where:

1) D⁰⁰is an e¤ective divisor of degreeusatisfyingD⁰⁰¼D⁰⁰, and havingP1;. . .;Prin its support.

2) D⁰is an e¤ective divisor such that there is no nonzero e¤ective divisorcD⁰invari- ant by the complex conjugation.

If D⁰¼0, then DADivðXÞ and r_CðX;rÞ ¼r_RðX;rÞ. So, assume D⁰00 and let l¼dimjD⁰⁰j. If r¼l, then jDj has a base point, but then XC has a complete g_k^r

with k<d, hence a contradiction. It follows that r>l. Since X is a real curve, dimCjD⁰⁰þD⁰j ¼r. We may find suitable nonzero e¤ective divisorsD₁⁰;. . .;D_rl⁰ such that

1) D⁰¼D₁⁰þ þD_rl⁰ , and

2) lCðD⁰⁰þD₁⁰þ þD_i⁰Þ ¼lCðD⁰⁰Þ þi,i¼1;. . .;rl.

Let f1;f1;. . .;frg be a base of H⁰ðXC;OðD⁰⁰þD⁰ÞÞ and g1;. . .;grlA

H⁰ðXC;OðD⁰⁰þD⁰ÞÞsuch that

1) g1AH⁰ðXC;OðD⁰⁰þD₁⁰ÞÞnH⁰ðXC;OðD⁰⁰ÞÞ, and

2) g_iAH⁰ðXC;OðD⁰⁰þD₁⁰þ þD_i⁰ÞÞnnH⁰ðXC;OðD⁰⁰þD₁⁰þ þD_i1⁰ ÞÞ, i¼ 2;. . .;rl.

Claim. lCðD⁰⁰þD⁰þD⁰Þd2rþ1l. More precisely, we show, by induction on i, that 1;f₁;. . .;f_r;g₁;. . .;g_i are linearly independent in the vector space H⁰ðXC;OðD⁰⁰þD₁⁰þ þD_i⁰þD⁰ÞÞ, i.e.lCðD⁰⁰þD₁⁰ þ þD_i⁰þD⁰Þdrþ1þi.

Fori¼1, 1;f₁;. . .;f_r,g₁AH⁰ðXC;OðD⁰⁰þD₁⁰þD⁰ÞÞand 1;f₁;. . .;f_rare linearly independent. If g₁ were a linear combination of 1;f₁;. . .;f_r, then g₁ would be a global section of OðD⁰⁰þD⁰Þand also divyðg1ÞcD⁰⁰þD⁰. By the construction of g₁, divyðg1ÞcD⁰⁰þD₁⁰. Since D⁰ and D₁⁰ have distinct supports, we would have divyðg1ÞcD⁰⁰. This is a contradiction.

Assume now that 1;f₁;. . .;f_r;g₁;. . .;gi1 (rl>i>1) are linearly independent and that g_i would be a linear combination of 1;f₁;. . .;f_r;g₁;. . .;gi1. Arguing as in the case i¼1, the pole divisor of gi would becD⁰⁰þD⁰þD₁⁰þ þD_i1⁰ . By the construction ofgi, and sinceD⁰andD_i⁰have distinct supports, we would obtain divyðgiÞcD⁰⁰þD₁⁰þ þD_i1⁰ , contradicting the fact that giBH⁰ðXC;OðD⁰⁰þ D₁⁰þ þD_i1⁰ ÞÞ. This ends the proof of the claim.

Since D⁰⁰þD⁰þD⁰ is invariant by the complex conjugation, we get lðD⁰⁰þD⁰þD⁰Þ ¼lCðD⁰⁰þD⁰þD⁰Þd2rþ1l. Let P₁⁰;. . .;P_rl⁰ be suitable real points. Then lðD⁰⁰þD⁰þD⁰P₁⁰ P_rl⁰ Þdrþ1, and X has at least one complete g_2durþl^r . To get the second assertion of the theorem, it is su‰cient to prove thatrcul. Since dim_CjP1þ þP_rj ¼0 andD⁰⁰¼P₁þ þP_rþE, with EADivðXÞan e¤ective divisor of degreeur, we getl¼dimCjP1þ þP_rþEjc

ur. r

Let us mention some consequences of Theorem 5.4.

Corollary 5.5. Let X be a real hyperelliptic curve and r be an integer such that 1c rcg1.Thenr_RðX;rÞ ¼2r.

Proof. Since r_CðX;rÞ ¼2r, using the first inequality of the theorem, the result fol-

lows. r

Corollary 5.6. Let X be a real curve of genus g which is not hyperelliptic, and r

be an integer such that 1cr<g1. Then r_RðX;rÞ>2r. In particular, we have r_Rð3;1Þ ¼3.

Proof. The proof is clear by Cli¤ord’s inequality and by the existence of real non-

hyperelliptic curves of genus 3. r

Corollary 5.7. Let X be a real curve. Then the mapj_g:ðS^gXÞðRÞ !Pic^gðXÞis not injective.

Proof.IfDADivðXÞof degreed satisfieslðDÞ ¼2, then the fiber ofj_d at½Dis one dimensional and the mapj_d is not injective. Theorem 5.4 asserts the existence of a

g_g¹onX, hence we get the result. r

Remark 5.8.In Theorem 5.4, we get two upper bounds forr_R, one of them depend- ing on r_C, but not the other. It is interesting to compare these two bounds. The invariantr_C is given by the Theorems 1.1 p. 206, 1.5 p. 214, in [1]. ForX a general curve of genusgand 1crcg1:

r_CðX;rÞ ¼gþr g rþ1

:

(½_rþ1^g is the integral part of_rþ1^g ). We thus obtain

r_RðX;rÞcmin gþr1;2g2 g rþ1

:

Assume_rþ1^g AN. We see that 2g2_rþ1^g ¼gþr1 ifr¼1 orr¼g1, and that 2g2_rþ1^g >gþr1 if not. Hence r_Rðg;rÞcgþr1 is the best upper bound we may find at this moment. As we have seen for hyperelliptic curves, the second inequality of Theorem 5.4 gives a smaller upper bound for r_RðX;rÞonly whenX is a special curve.

5.2 Complete linear systems of dimension 1 on real curves.The following theorem is a refinement of Theorem 5.4 in dimension 1.

Proposition 5.9.Let X be a real curve of genus gd2.If XChas exactly an odd number of g¹_r

CðX;1Þ,thenr_RðX;1Þ ¼r_CðX;1Þ.

Proof.We have r_CðX;1Þd2 sinceX is not rational. Letd ¼r_CðX;1Þ. Assume that XC has exactly 2nþ1 distinct g_d¹, nAN. Let D_i⁰, i¼1;. . .;2nþ1, some e¤ective divisors on XC such that the linear systemsjD_i⁰j are the 2nþ1 distinctg_d¹. We may clearly assume that, for everyi,D_i⁰¼PþD_i, withPa closed point ofXC satisfying P¼Pand withD_iADivðXCÞan e¤ective divisor of degreed1. SinceX is real, the linear systems jD_i⁰j are also g_d¹. Consequently, there exists kAf1;. . .;2nþ1g such that jD_k⁰j ¼ jD_k⁰j, since an involution acting on a finite set with an odd number of elements has a fixed point. Hence PþD_k is linearly equivalent to PþD_k. Conse-

quently, either D_k ¼D_k and we have the claim, orjDkj is ag_d1¹ , but thend is not

minimal. r

Corollary 5.10. Let X be a general real curve of genus 6. Then r_CðX;1Þ ¼ r_RðX;1Þ ¼4.

Proof.Since XC has fiveg₄¹(see [3] p. 299), the proof follows from the above propo-

sition. r

5.3 A real Brill–Noether number. In the context, a natural question one can ask is about the existence of a real curve X of genusgd2 withr_RðX;1Þ ¼g. Such an existence, for anygd2, would show thatr_Rðg;1Þ ¼g.

If d<g and X is a real curve of genus g having a g_d¹, then, adding ðg1dÞ general real points to this g_d¹, we get a completeg_g1¹ . By [1] Corollary 4.5, p. 190, the singularities of Wg1¼j_g1ðS^g1XCÞ are the complete g_g1^k with k>0, where

j_g1:S^g1XC!Pic^g1ðXCÞ; ðP1;. . .;Pg1Þ 7! ½P1þ þPg1 is the natural map.

By Riemann’s singularity theorem (see [1] p. 226), the singular part of the theta divisor yJJðCÞ is a translation (by a theta-characteristic) of the singular part of Wg1. Recall that a real curve always admits real theta-characteristics [4]. Hence we may reformulate the previous question asking if there exist real curves of genusgd2 withyðRÞnon-singular. We state the following conjecture:

Conjecture 1.Let gd2be an integer.There exists a real curve X of genus g such that the singularities of the theta divisoryJJðCÞare not real.

Proposition 5.11.The above conjecture holds for2cgc4.

Proof.The conjecture holds trivially for genus 2 curves, and genus 3 curves since there exist non-hyperelliptic real curves of genus 3.

Following Gross and Harris [4], we may show that the conjecture holds for genus 4 curves. Let X be a real trigonal curve (i.e. r_CðX;1Þc3) of genus 4 which is non- hyperelliptic. Its canonical model lies on a unique real quadric surface SJP³

R. For a generalX,Sis smooth and then has two di¤erent rulings. For someX, these two rulings are complex and switched by the complex conjugation. Then XC has only two g₃¹ induced by these two rulings and r_CðX;1Þ ¼3. By this, we conclude that

r_RðX;1Þ ¼2r_CðX;1Þ 2¼4. r

Letg;rANsatisfyinggd2 and 1crcg1. Ifd ðgþr1Þd0, then Theo- rem 5.4 says that any real curve of genusghas a completeg_d^r. We may wonder if this condition is optimal.

Conjecture 2. Let gd2 and 1crcg1. The real Brill–Noether number is r_Rðg;r;dÞ ¼d ðgþr1Þ,i:e:if d ðgþr1Þ<0,then there exists a real curve X of genus g such that X has no g_d^r.

Remark that Conjecture 2 implies Conjecture 1.

6 Linear systems with base points on real curves

This section is devoted to the problem of finding lower bounds forN. We prove that this problem is related to the existence of special linear systems of dimension 1 and small degree, i.e. the subject of the previous section.

Proposition 6.1. Let X be a real curve of genus gd2.Assume that X has a complete g_d^r,with dcg1and rd1.Then NðXÞ>2gd.

Proof.LetDADivðXÞbe an e¤ective divisor of degreedcg1 such that dimjDj ¼ rd1. It is a special divisor. Let D⁰ADivðXÞ be an e¤ective divisor of degree 2g2d such thatDþD⁰is the canonical divisor. LetQbe a non-real point ofX such thatlðDQÞ ¼lðDÞ 2. By Riemann–Roch,

lðD⁰þQÞ ¼2g2dþ2gþ1þlðDQÞ ¼gdþ1þr1¼lðD⁰Þ: Hence Qis a base point of jD⁰þQjand consequently the divisorD⁰þQof degree 2gd is not linearly equivalent to a totally real e¤ective divisor. Clearly NðXÞ>

2gd. r

The existence of linear systems of small degree on real curves is studied in the pre- vious section. One of the results, is that a real curve has always a completeg_g¹, hence Corollary 6.2.Let X be a real curve such that gd2.Then NðXÞdgþ1.

From the previous section and Proposition 6.1, we obtain

Corollary 6.3.Let X be a real curve of genus gd2.IfyJJðCÞhas a real singularity, then NðXÞdgþ2.

IfXis hyperelliptic, by Lemma 4.2,Xhas also ag₂¹and we may state the following result (use Theorem 3.6):

Corollary 6.4.Let X be a real hyperelliptic curve of genus gd2.Then NðXÞd2g1.

If furthermore X is an M-curve or anðM1Þ-curve,then NðXÞ ¼2g1.

Remark 6.5.Since there exist real hyperelliptic M-curves of any genus, the previous corollary gives a large family of curves for which the invariant N is explicitely cal- culated.

References

[1] E. Arbarello, M. Cornalba, P. A. Gri‰ths, J. Harris,Geometry of algebraic curves. Vol. I.

Springer 1985. MR 86h:14019 Zbl 0559.14017

[2] J. Bochnak, M. Coste, M.-F. Roy,Ge´ome´trie alge´brique re´elle. Springer 1987.

MR 90b:14030 Zbl 0633.14016

[3] P. Gri‰ths, J. Harris,Principles of algebraic geometry. Wiley-Interscience 1978.

MR 80b:14001 Zbl 0408.14001

[4] B. H. Gross, J. Harris, Real algebraic curves.Ann. Sci. E´cole Norm. Sup.(4)14(1981), 157–182. MR 83a:14028 Zbl 0533.14011

[5] R. Hartshorne,Algebraic geometry. Springer 1977. MR 57 #3116 Zbl 0367.14001 [6] J. Huisman, On the geometry of algebraic curves having many real components. Rev.

Mat. Complut.14(2001), 83–92. MR 2002f:14074 Zbl 1001.14021

[7] J. Huisman, On the neutral component of the Jacobian of a real algebraic curve having many components.Indag. Math.(N.S.)12(2001), 73–81. MR 1 908 140 Zbl 01663809 [8] J. Huisman, Cli¤ord’s inequality for real algebraic curves. Preprint 2000.

[9] J. Huisman, Non-special divisors on real algebraic curves and embedding into real pro- jective spaces. Preprint 2000.

[10] S. L. Kleiman, D. Laksov, Another proof of the existence of special divisors.Acta Math.

132(1974), 163–176. MR 50 #9866 Zbl 0286.14005

[11] C. Scheiderer, Sums of squares of regular functions on real algebraic varieties. Trans.

Amer. Math. Soc.352(2000), 1039–1069. MR 2000j:14090 Zbl 0941.14024

Received 24 April, 2002; revised 4 October, 2002 and 11 December, 2002

J.-Ph. Monnier, De´partement de Mathe´matiques, Universite´ d’Angers, 2, Bd. Lavoisier, 49045 Angers cedex 01, France

Email: monnier@tonton.univ-angers.fr