# Very ample linear series on real algebraic curves.

1 Introduction and Terminology

Let Y be a complex algebraic curve, i.e. a smooth and irreducible projective curve defined over C, and let g be its genus. The distribution of complete and very ample linear series [g.sup.r.sub.d] of degree d and (projective) dimension r on Y is only known if they are not very special, i.e. if [h.sup.1] (Y, D) [less than or equal to] 1 holds for the index of speciality of a divisor D [member of] [g.sup.r.sub.d] : Y always has complete, very ample and non-special series of dimension r for all integers r [greater than or equal to] 3, and there is no complete, very ample and special series on Y if and only if Y is hyperelliptic. More precisely, Y has a complete, very ample, special but not very special linear series of dimension r (i.e. a complete [g.sup.r.sub.g-1+r]) for every integer r such that 3 [less than or equal to] r < g provided that Y is not hyperelliptic, not bi-elliptic (i.e. not a double cover of an elliptic curve), not trigonal and not a smooth plane quintic; this statement is a consequence of Mumford's dimension theorem for the varieties of special divisors. (Cf. [ACGH], V, Ex. B; [CM1], 2.2 (iv); by the way, for g [greater or equal to] 6 also the converse of this statement is true). It is the aim of this paper to prove the corresponding very ampleness-results for complete linear series on a real algebraic curve, i.e. on a smooth and geometrically irreducible projective curve defined over R. We always assume that the real curve has real points. (On a real curve without real points linear series are restricted to even degree.)

Terminology: For details on real curves and the language of linear series on them we refer to [CM 2]. Linear series are always complete in this paper. X denotes a real (algebraic) curve of genus g, and ([X.sub.C], [sigma]) denotes the complexification of X([sigma] is the anti-analytic involution on [X.sub.C] induced by complex conjugation--on C). If we assume that the set X(R) of real points of X (or, what is the same, of [X.sub.C]) is non-empty it splits into 1 [less than or equal to] s [less than or equal to] g + 1 real components [C.sub.1], ..., [C.sub.s] of X, and we can identify the linear series [g.sup.r.sub.d] on the real curve X with the ([sigma]-invariant [g.sup.r.sub.d] on the complex curve [X.sub.C] ([GH], section 2; [CM2], section 1). Such a "real" [g.sup.r.sub.d] on [X.sub.C] defines, for r [greater than or equal to] 1, a rational map [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] which is [sigma]-invariant (i.e. <[phi]([P.sup.[sigma]]) = [bar.[phi](P)] for any point P [member of] [X.sub.C] where it is defined; this is due to the fact that [H.sup.0] ([X.sub.C], D) for D [member of] [g.sup.r.sub.d] has a basis of real functions because [([g.sup.r.sub.d]).sup.[sigma]] = [g.sup.r.sub.d]); so [phi] induces a map [phi] : X [right arrow] [P.sup.r.sub.R] = Proj(R[[x.sub.0], ..., [x.sub.r]]).

A [g.sup.r.sub.d] on X is called very ample if the associated [sigma]-invariant [g.sup.r.sub.d] on [X.sub.C] is very ample, i.e. if the induced map [phi] is an isomorphism onto the image curve in [P.sup.r.sub.C]; then [phi] identifies the real curve X with a real curve in the real projective space [P.sup.r.sub.R]. The very ample [g.sup.r.sub.d] on X correspond to the elements of the subset [W.sup.r.sub.d](R)\([W.sup.r-1.sub.d-2] + [W.sub.2])(R) of the Jacobian variety Jac([X.sub.C]) of [X.sub.C]; here [W.sup.r.sub.d] resp. [W.sup.r.sub.d](R) represents the set of [g.sup.r.sub.d] resp. of [sigma]-invariant [g.sup.r.sub.d] on [X.sub.C]. A linear series [g.sup.r.sub.d] on X belonging to the-in general larger-set [W.sup.r.sub.d](R)\([W.sup.r-1.sub.d-2](R) + [W.sub.2](R)) still provides a nice geometric description of X: it is easy to see that such a series is base point free and simple (i.e. [phi] is a birational morphism from [X.sub.C] onto its image; so r [greater than or equal to] 2), and it has the defining two properties that the morphism [phi] separates conjugate points of [X.sub.C] (i.e. [phi]([P.sup.[sigma]]) [not equal to] [phi](P) for every non-real point P of [X.sub.C]; in particular, the image curve [phi](X) [subset] [P.sup.r.sub.R] has no isolated real points) and separates points and tangent vectors on X(R) (so [X.sub.C] has its singular points outside X(R); in particular, the real contours [phi]([C.sub.1]), ..., [phi]([C.sub.s]) of [phi]([X.sub.C]) do not meet).

Note that these notions are also meaningful for a real curve without real points; in this paper, however, we assume that X(R) [not equal to] [phi] (i.e. s [greater than or equal to] 1).

Example V. There are real curves of genus 4 with a given number s, 1 [less than or equal to] s [less than or equal to] 5, of real components and without a [g.sup.1.sub.3]([GH], p. 178). Let X be such a curve and let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] denote the canonical series of [X.sub.C]. Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is [sigma]-invariant so is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for any P [member of] [X.sub.C](R) = X(R); thus the [[infinity].sup.1] nets [g.sup.2.sub.5] on X belong to ([W.sup.1.sub.3] + [W.sub.2])(R) but clearly not to [W.sup.1.sup.3](R)+[W.sub.2](P) = [phi].

A pseudo-line for a [g.sup.r.sub.d] (r [greater than or equal to] 1) on X is a real component of X on which some (hence every) divisor in this [g.sup.r.sub.d] has odd degree. The number [delta] of pseudo-lines for a [g.sup.r.sub.d] on X obviously satisfies 0 [less than or equal to] [delta] [less than or equal to] Min (s, d) and [delta] [equivalent to] d mod 2, and if the [g.sup.r.sub.d] is special (i.e. r > d - g) then [delta] [less than or equal to] 2g - 2 - d ([H]). In particular, the canonical series [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] of [X.sub.C] has no pseudo-lines.

2 Non-special very ample series

We identify the real points of X with the [sigma]-invariant points of [X.sub.C]. Recall that X has s [less than or equal to] 1 real components.

Lemma 1: Let d, 3 be non-negative integers such that [delta] [less than or equal to] Min(s, d) and [delta] [equivalent to] d mod 2. Choose [delta] general points [P.sub.1], ..., [P.sub.[delta]] on real components [C.sub.1], ..., [C.sub.[delta]] of X, respectively, and d-[delta]/2 general points [Q.sub.1], ..., [Q.sub.d-[delta]/2] in [X.sub.C]. Let D be the real (= [sigma]-invariant) and effective divisor [P.sub.1] + x x x + [P.sub.[delta]] + ([Q.sub.1] + [Q.sup.[sigma]/sub.1]) + x x x + ([Q.sub.d-[delta]/2] + [Q.sup.[sigma].sub.d-[delta]/2]) of degree d on [X.sub.C]. Then dim [absolute value of D] =Max(0,rf - g).

Proof: Clearly, by Riemann-Roch, r := dim [absolute value of D] [greater than or equal to] Max (0,d - g). Our real divisors D of degree d on [X.sub.C] form a family of real dimension at least d. Taking the complete linear series they define we see that [dim.sub.R]([W.sup.r.sub.d](R)) [greater than or equal to] d - r. Let d < g. Then Max(0,rf - g) = 0, and we have d-r < [dim.sub.R](W.sup.r.sub.](R)) [less than or equal to] dim ([W.sup.r.sub.d]) [less than or equal to] d-2r, by Martens' dimension theorem ([ACGH], IV, 5.1). Hence r = 0, and we are done.

Let d > [greater than or equal to], i.e. Max(0, d - g) = d - g. Assume that r > d - g. Then r > 0. Let d' := 2g - 2 - d , r' := g - 1 - d + r [greater than or equal to] 0. By Martens' dimension theorem, d-r [less than or equal to] [dim.sub.R]([W.sup.r.sub.r](R)) [less than or equal to] dim([W.sup.r.sub.d]) = dim([kappa] - [W.sup.r'.sub.d']) = dim([W.sup.r'.sub.d']) [less than or equal to] d! - 2r' = d - 2r, again; here {[kappa]} = [W.sup.g-1.sub.1g-2] is the canonical point on Jac([X.sub.C]) corresponding to the canonical series [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. It follows that r < 0, a contradiction.

Proposition 1: Let r [greater than or equal to] 3 and [delta] [greater than or equal to] 0 be integers such that [delta] [less than or equal to] s and [delta] [equivalent to] g + r mod 2. Then for [delta] assigned real components [C.sub.1], ..., [C.sub.[delta]] of X there is a complete and very ample [g.sup.r.sub.g+r] on X having precisely the pseudo-lines [C.sub.1], ..., [C.sub.[delta]].

Proof. The proof is just a straightforward modification of the proof of [CM2], Proposition 1. Choose D as in the lemma, with d :=g + r;then dim[absolute value of D] = d - g = r. For every real divisor D' [member of] [absolute value of D] and every real component C of X we have deg(D'[|.sub.C]) [equivalent to] deg(D[|.sub.C]) mod 2; so, by construction of D, precisely [C.sub.1], ..., [C.sub.[delta]] are the pseudo-lines for [absolute value of D].

Let r [greater than or equal to] 3 and assume that no such [absolute value of D] is very ample. Then these series constitute a d - r = g-dimensional subset of Jac([X.sub.C])(R) contained in ([W.sup.r-1.sub.g+r-2] + [W.sub.2]) (R). Hence [dim.sub.R]([W.sup.r-1.sub.g+r-2] + [W.sub.2])(R)) [greater than or equal to] g. But [W.sup.r-1.sub.g+r-2] = k - [W.sub.g-r] has dimension g - r (if r [less than or equal to] g; otherwise it is empty), and so we have [dim.sub.R](([W.sup.r-1.sub.g+r-2]+[W.sub.2]) (R)) [less than or equal to] dim([W.sup.r-1.sub.g+r-2] + [W.sub.2]) < (g - r) + 2 < g, a contradiction.

For r = 2 we can, of course, only expect a weaker result:

Proposition 2: Let [delta] [greater than or equal to] 0 be an integer such that [delta] [less than or equal to] s and [delta] [equivalent to] g mod 2. Then for [delta] assigned real components [C.sub.1], ..., [C.sub.[delta]] of X there is a complete, base point free and simple net [g.sup.2.sub.g+2] on X having precisely the pseudo-lines [C.sub.1], ..., [C.sub.[delta]]. And if X is not hyperelliptic (i.e. has no [g.sup.1.sub.2]) the real plane curve obtained by this net has only points of multiplicity at most 2.

Proof: Let [W.sup.1.sub.g] x [W.sub.2] [right arrow] Tac([X.sub.C]) be the summation map and W(g,1) be the closed and ([sigma]-invariant sublocus of Jac([X.sub.C]) consisting of points whose fibre under this map is not finite. Then Jac([X.sub.C])(R)\W(g, 1)(R) collects the complete, base point free and simple nets [g.sup.2.sub.g+2] on X. Since [dim.sub.R](W(g, 1)(R)) [less than or equal to] dim(W(g/1)) < dim([W.sup.1.sub.g] x [W.sub.2]) = dim([W.sup.1.sub.g+2]) + 2 = dim([kappa] - [W.sub.g-2]) + 2 = g we see that the g-dimensional subset of Jac([X.sub.C])(R) made up by the series [absolute value of D] constructed as in the proof of Proposition 1 cannot be contained in W(g, 1) (R).

Likewise, if the image curve of the morphism defined by [absolute value of D] has a singular point of multiplicity m > 3 than [absolute value of D] represents a point in ([W.sup.1.sub.g+2-m] + [W.sub.m])(R). But, by Martens' dimension theorem ([ACGH], IV, 5.1), dim([W.sup.1.sub.g+2-m]) [less than or equal to] g - 1 - m if [X.sub.C] is not hyperelliptic, and then dim([W.sup.1.sub.g+2-m] + [W.sub.m]) [less than or equal to] g - 1 only.

Example 2: Let [delta] = g (maximal) in Proposition 2. By [H], 2.7, then, the [g.sup.2.sub.g+2] on X separates conjugate points, and its induced morphism [phi] restricts to an isomorphism on every real components of X. Let [[GAMMA].sub.1], ..., [[GAMMA].sub.g] be the images (under [phi]) of the g pseudo-lines [C.sub.1], ..., [C.sub.g] of X. Since any two of the [[GAMMA].sub.i] intersect in [P.sup.2.sub.C](R) we have at least ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]) such intersection points, and since the singular plane curve [phi]([X.sub.C]) cannot have more than [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] singular points we see that [phi](X) [subset] [P.sup.2.sub.R] has precisely the [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] points of intersection of the real contours [[GAMMA].sub.i] as its singularities.

Let g [greater than or equal to] 2. Since [[GAMMA].sub.i] [intersestion] [[GAMMA].sub.i] [not equal to] for i [not equal to] j there are points P [member of] [C.sub.i], Q [member of] [C.sub.j] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Remarks: (i) For r [greater than or equal to] 3 let [W.sup.r-2.sub.g-4+r] x [W.sub.4] [right arrow] Jac([X.sub.C]) be the summation map and W(g - 4 + r, r - 2) be the closed and [sigma]-invariant sublocus of Jac([X.sub.C]) consisting of points whose fibre under this map is not finite. Assume that X is not hyperelliptic. If, then, the very ample series [g.sup.r.sub.g+r] found in Proposition 1 would admit infinitely many m-secant line divisors for some m [greater than or equal to] 4 (i.e. if the curve of degree g + r in [P.sup.r] embedded by our [g.sup.r.sub.g+r] would have [[infinity].sup.1] quadrisecant lines) we would have [dim.sub.R](W(g - 4 + r,r - 2) (R)) [greater than or equal to] g which implies g [less than or equal to] dim(W(g - 4 + r,r - 2) < dim([W.sup.r-2.sub.g-4+r] x [W.sub.4]) = dim([W.sup.r-2.sub.g-4+r]) + 4 [less than or equal to] ((g - 4 + r) - 2(r - 2) - 1) + 4 = g - r + 3, by Martens' dimension theorem; so r [less than or equal to] 2. This contradiction shows that our [g.sup.r.sub.g+r] has only a finite number of m-secant line divisors (m [greater than or equal to] 4).

(ii) The result in (i) resp. the last statement in Proposition 2 (on singularities) are false for hyperelliptic X since [g.sup.r.sub.g+r] = [absolute value of [g.sup.r.sub.2] + [g.sup.r-2.sub.g+r-2]] then.

3 Special very ample series

The following lemma is well-known and due to the fact that a reduced and irreducible complex curve in [P.sup.r](r [greater than or equal to] 3) cannot have [[infinity].sup.1] singular points resp. [[infinity].sup.2] trisecant lines ([ACGH], III, ex. L).

Lemma 2: Let L = [g.sup.r.sub.d] (r [greater than or equal to] 3) be a base point free and simple linear series on the complex curve [X.sub.C]. Then there are only finitely many points P [member of] [X.sub.C] such that [g.sup.r-1.sub.d-z] := [absolute value of L - P] is not both base point free and simple.

The lemma implies

Proposition 3: Let X be non-hyper elliptic and 2 [less than or equal to] r < g and [delta] [greater than or equal to] 0 be integers such that [delta[] < Min(s,g - 1 - r) and [delta] [??] g - r mod 2. Then for assigned [delta] real components [C.sub.1], ..., [C.sub.[delta]] of X there is a complete, base point free and simple [g.sup.r.sub.g-1+r] on X having precisely the pseudo-lines [C.sub.1], ..., [C.sub.[delta]].

Proof: Note that the conditions on 3 are necessary for [delta] pseudo-lines for [delta] (complete and special) [g.sup.r.sub.g-1+r] on X.

Since X is not hyperelliptic so is its complexification [X.sub.C]. Consequently, [X.sub.C] has a canonical divisor [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] which is real and very ample, and from section 1 we know that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] has no pseudo-lines. This proves the Proposition for r = g - 1. Assume that the Proposition is true for some 3 [less than or equal to] r < g; we want to prove it for r - 1 by using Lemma 2, i.e. we want to show the existence of a base point free and simple [g.sup.r-1.sub.g-2+r] on X with [delta] assigned pseudo-lines [C.sub.1], ..., [C.sub.[delta]] provided that 0 [less than or equal to] [delta] [less than or equal to] Min(s,g - r) and [delta] [equivalent] g - r mod 2.

If [delta] > 0 take a base point free and simple [g.sup.r.sub.g-1+r] on X having exactly the [delta] - 1 pseudo-lines [C.sub.1], ..., [C.sub.i]; it represents a (T-invariant series on [X.sub.C]. By Lemma 2 we can choose a point [P.sub.[delta]] [member of] [C.sub.[delta]] such that [absolute value of [g.sup.r.sub.g-1+r] - [P.sub.[delta]] is a [sigma]-invariant, base point free and simple [g.sup.r-1.sub.g-2+r] on [X.sub.C]. Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] has the pseudo-lines [C.sub.i], ..., [C.sub.[delta]] the series [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] hence also its dual [absolute value of [g.sup.r.sub.g-1+r] - [P.sub.[delta]] have the pseudo-lines [C.sub.1], ..., [C.sub.[delta]]

If [delta] = 0 (note that this implies r < g - 1) we take a base point free and simple [g.sup.r.sub.g-1+r] with exactly one pseudo-line C, and by Lemma 2 we can find a point P [member of] C such that (on [X.sub.C]) [absolute value of [g.sup.r.sub.g-1+r -P]] is base point free and simple, again. Since this series has even degree on C we see that it has no pseudo-line.

Corollary: Let g [greater than or equal to] 5 and s [greater than or equal to] 3 (s [greater than or equal to] 2 suffices for odd g). Then [W.sup.1.sub.g-1] (R) [not equal to] [phi].

Proof: If X is hyperelliptic we have [W.sup.1.sub.2](R) [not equal to] [phi], and so [W.sup.1.sub.2] (R) + [W.sub.1](R)+ x x x + [W.sub.1](R) (with g - 3 varieties [W.sub.1](R)) is contained in [W.sup.1.sub.g-1](R), for g [greater than or equal to] 3. If X is not hyperelliptic we apply Proposition 3 with r = 2 and [delta] = 2 for odd g resp. ([delta] = 3 for even g. Then the image curve in [P.sup.2.sub.C] under the birational morphism induced by the chosen net [g.sup.2.sub.g+1] on [X.sub.C] has real singular points (the points of intersection of the images of the [delta] [greater than or equal to] 2 pseudo-lines), and the projection off such a singularity onto [P.sup.1.sub.C] given us a (T-invariant pencil of degree at most g - 1 on [X.sub.C].

Finally, for r [greater than or equal to] 3 we have the

Theorem: Assume that X has no [g.sup.sub.3], is not a smooth plane quintic and not a double cover of a real elliptic curve. Let 3 [less than or equal to] r < g and [delta] [greater than or equal to] 0 be integers such that [delta] [less than or equal to] Min(s, g - l - r) and [delta] [??] g - r mod 2. Then for 5 assigned real components [C.sub.1], ..., [C.sub.[delta]] of X there is a complete and very ample [g.sup.r.sub.g-1+r] on X having precisely [C.sub.1], ..., [C.sub.[delta]] as its pseudo-lines.

Proof: The inequality 3 [less than or equal to] r < g implies g [greater than or equal to] 4, and for g = 4 we have r = 3, i.e. [g.sup.r.sub.g-1+r] is the canonical series on X, and so [delta] = 0. Hence we may assume that g [greater than or equal to] 5. Choose D as in Lemma 1, with d = g - 1 - r [greater than or equal to] 0 there. Then dim [absolute value of D] = 0, and the series [absolute value of D] form a subset of real dimension g - 1 - r in [W.sub.g-1-r](R). Since [X.sub.C] has a real canonical divisor [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] the series [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII complete and [sigma]-invariant [g.sup.r.sub.g-1+r] on [X.sub.C] with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], and since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] mod 2 for every real component C of X we see that precisely the 5 assigned real components [C.sub.1], ..., [C.sub.[delta]] of X are the pseudo-lines for these [g.sup.r.sub.g-1+r].

The series [g.sup.r.sub.g-1+r] thus constructed constitute a (g - 1 - r)-dimensional subset Z of [W.sup.r.sub.g-1+r](R) = ([kappa] - [W.sub.g-1-r])(R) = [kappa] - [W.sub.g-1-r](R). Assume that none of these [g.sup.r.sub.g-1+r] is very ample, i.e. Z [subset or equal to] ([W.sup.r.sub.g-3+r]+[W.sub.2])(R) = (([kappa] - [W.sup.1.sub.g-1+r]) + [W.sub.2])(R) = [kappa] - ([W.sup.r.sub.g-1+r] - [W.sub.2])(R). Then Z' := [kappa] - Z is a (g - 1 - r)-dimensional subset of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Claim 1: dim([W.sup.1.sub.g-r+r] [less than or equal to] g - 3 - r for r [greater than or equal to] 3.

In fact, the claim is clear provided that the complex curve [X.sub.C] is not hyperelliptic, not trigonal, not bi-elliptic and not a smooth plane quintic, according to Mumford's dimension theorem ([ACGH], IV, 5.2). Assume that [X.sub.C] is hyperelliptic or trigonal. Then [X.sub.C] has a unique [g.sup.1.sub.2] resp. a unique [g.sup.1.sub.3] (recall g [greater than or equal to] 5) which - being unique - must be [sigma]-invariant; so X has a [g.sup.1.sub.3] which contradicts our hypotheses. If [X.sub.C] is bi-elliptic and g >[greater than or equal to] 6 the covered elliptic curve Ec is unique ([ACGH], VIII, Ex. C - 1). Then a moves [E.sub.C] into itself whence there is a real elliptic curve E doubly covered by X whose complexification is [E.sub.C]. This again contradicts our hypotheses. If [X.sub.C] is bi-elliptic and g = 5 we have r = 3 or r = 4 which implies [W.sup.1.sub.g-1+r] = [phi] ([ACGH], VIII, Ex. C -1). Finally, if [X.sub.C] is a smooth plane quintic (g = 6) it has a unique net [g.sup.2.sub.5] which (being unique) must be [sigma]-invariant. Then X is a smooth real plane quintic, a case we have excluded. This proves the claim.

Claim 2: [W.sub.g-1-r] [??] [W.sup.1.sub.g-r+1] - [W.sub.2].

In fact, claim 1 yields [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] [W.sub.2] would imply that there is an irreducible component Y of [W.sup.1.sub.g-1+r] of dimension g - 3 - r such that [W.sub.g-1-r]= Y - [W.sub.2]. But since Y [??] [W.sup.2.sub.g-r+1] we have Y - [W.sub.2] [??] [W.sub.g-1-r].

Claim 2 implies that [W.sup.1.sub.g-1-r] [intersection] ([W.sup.1.sub.g-1-r] - [W.sub.2]) is properly contained in the irreducible variety [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] This contradiction shows that our assumption Z [subset or equal to] ([W.sup.r-1.sub.g-3+r] + [W.sub.2]) (R) is false.

Received by the editors November 2009.

References

[ACGH] Arbarello, E.; Cornalba, M.; Griffiths, P.A.; Harris, J.: Geometry of algebraic curves. Vol. I. Grundlehren 267 (1985), Springer Verlag

[CM1] Coppens, M.; Martens, G.: Linear series on 4-gonal curves. Math. Nachr. 213 (2000), 35-55

[CM2] Coppens, M.; Martens, G.: Linear pencils on real algebraic curves. Preprint (2008)

[GH] Gross, B.H.; Harris, J.: Real algebraic curves. Ann. scient. Ec. Norm. Sup., 4. ser., 14 (1981), 157 - 182

[H] Huismann, J.: Non-special divisors on real algebraic curves and embeddings into real projective spaces. Ann. di Mat. 182 (2003), 21-35

* This research is partially supported by the Fund of Scientific Research-Flanders (G. 0318.06)

Marc Coppens

Katholieke Industriele Hogeschool der Kempen, Campus HI Kempen,

Kleinhoefstraat 4, B-2440 Geel, Belgium

and: K. U. Leuven, Department of Mathematics, Celestijnenlaan 200 B,

B-3001 Heverlee, Belgium

marc.coppens@khk.be

Gerriet Martens

Universitat Erlangen-Nurnberg, Department Mathematik,

Bereich Algebra und Geometrie, Bismarckstr. 11/2, D-91054 Erlangen, Germany

martens@mi.uni-erlangen.de
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