Marc Coppens

Primitive linear series on curves

In this paper we study a new numerical invariant ℓ of curvesC which is related to the primitive linear series onC. (Primitive series—defined below—are the essential complete and special linear series onC.) The curves with ℓ≤3 are classified, and it is shown that for a given value of ℓ the curve is a double covering if its genus is sufficiently high. The main tool are dimension theorems of H. Martens-Mumford-type for the varieties of special divisors ofC, and we prove two refinements of these theorems.

The gonality of smooth curves with plane models

LetC be the normalization of an integral plane curve of degreed with δ ordinary nodes or cusps as its singularities. If δ=0, then Namba proved that there is no linear seriesgd−2/1 and that everygd−1/1 is cut out by a pencil of lines passing through a point onC. The main purpose of this paper is to generalize his result to the case δ>0. A typical one is as follows: Ifd≥2(k+1), and δ<kd−(k+1)2+3 for somek>0, thenC has no linear seriesgd−3/1. We also show that ifd≥2k+3 and δ<kd−(k+1)2+2, then each linear seriesgd−2/1 onC is cut out by a pencil of lines. We have similar results forgd−1/1 andg2d−9/1. Furthermore, we also show that all of our theorems are sharp.

Linear Series on 4 – Gonal Curves

Let C be the general 4 – gonal curve of genus g > 6. We investigate the complete linear series on C and the varieties Wrd(C), and we study the birational models of C in ℙr (r ≥ 2) of minimal degree.

Linear series on a general k-gonal curve

This paper is a contribution towards a Brill-Noether theory for the moduli space of smooth &-gonal curves of genusg. Specifically, we prove the existence of certain special divisors on a generalk-gonal curveC of genusg, and we detect an irreducible component of the “expected” dimension in the varietyWrd(C), (r ≤k — 2) of special divisors ofC. The latter induces a new proof of the existence theorem for special divisors on a smooth curve.

Free linear systems on integral Gorenstein curves

In the study of linear systems on smooth curves, it is enough to consider base point free linear systems. Indeed, each linear system is obtained by adding some base points to a uniquely determined base point free linear system. In [S], R. Hartshorne defined complete linear systems on integral Gorenstein curves. In [6, Example 1.6.11 he shows that the base point free linear systems are not that essential any more. In this paper, we introduce the notion of free linear systems on integral Gorenstein curves. We obtain that each linear system on such a curve is obtained by enlarging the base scheme (see Definition 2.1.2) of a uniquely determined free linear system. Singular points on the curve can appear as base points of free linear systems. We introduce the notion of base-point- degree of a linear system at a point. We prove that, in the case of free linear systems, this number is bounded from above by the kind of singularity. Moreover, let r be an integral curve on a smooth surface X and let P be a linear system on X. P induces a linear system on L’. We find upper bounds for the base-point-degree of this linear system in terms of intersection multiplicities. A very useful lemma in the study of linear systems on smooth curves is the base point free pencil trick (see, e.g., [ 1, p. 1261). In our context, we prove a free pencil trick. This is used in order to prove a result about free pencils on integral plane curves. In my paper [3], this result is essential for the study of linear systems on smooth plane curves. As an application; we show that, if C is the normalization of an integral plane curve f of degree d> 12 with 6 <

Some remarks on the schemesW d r

SummaryLet X be an irreducible smooth projective curve of genus g. Let ϱdr(g) be the Brill-Noether Number. In this paper we prove some results concerning the schemes Wdr of special divisors. 1) Suppose dim (Wd−1r)=ϱd− 1r(g)⩾0 and ϱdr(g) < g. If Wd− 1r is a reduced (resp. irreducible) scheme, then Wdr is a reduced (resp. irreducible) scheme. 2) Under certain conditions, if Z is a generically reduced irreducible component of Wd−1r then Z ⊕ W10 is a generically reduced irreducible component of Wdr. For r=1, we obtain some further results in this direction. 3) As an application of it we are able to prove some dimension theorems for the schemes Wd1.

An infinitesimal study of secant space divisors

For a linear system gf we define the subscheme vi-f of the symmetric product C@) parametrizing e-secant (e -f- 1)-space-divisors. For an element E of Vzlee-5 we study the Zariski-tangent space. We give a description of this tangent space in terms of intersections with hyperplanes and quadrics. In particular, we are able to give an easy formula for the dimension of that tangent space for the case of secant line divisors. This completely solves a problem studied in our earlier paper [3]. In the appendix some of the results in Coppens’ article are given natural interpretations in terms of local properties of subschemes of Grassmannians, such that these subschemes parametrize e-secant (e -f- 1)-planes.

On the varieties of special divisors

Abstract Based on a relation between the varieties W d r ( C ) of special divisors on a curve C and subloci of effective divisors on C imposing a suitable number of conditions on a certain linear series we develop a tool for the construction of irreducible components of W d r ( C ). Using this we discover new irreducible components of W d r ( C ), for a general k -gonal curve C of genus g , and in some cases we can identify the duals of these components in K C − W d r ( C ) = W d ′ r ′ ( C )( d ′ = 2 g − 2 − d , r ′ = g − 1 − d + r ).

Brill–Noether Theory for Non-Special Linear Systems II: Connectedness and Irreducibility

Let C be a general smooth curve of genus g and let gdnbe a general non-special linear system on C. Under some numerical conditions we prove the irreducibility of the scheme parametrizing secant space divisors for gdn. We also prove connectedness under some weaker conditions.

A metric graph satisfying [...] w 4 1 = 1

Abstract For all integers g ≥ 6 we prove the existence of a metric graph G with w41=1