Gerriet Martens

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.

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.

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 ).

Minimal genus of Klein surfaces admitting an automorphism of a given order

Let K be a compact Klein surface of algebraic genus

Algebraic curves violating the slope inequalities

g\ge 2,

Real line bundles on k-gonal real curves

which is not a classical Riemann surface. The authors show that if K admits an automorphism of order

Linear Series Computing the Clifford Index of a Projective Curve

N>2,

Secant spaces and Clifford's theorem

then it must have algebraic genus at least

The gonality of curves on a Hirzebruch surface

(p\sb 1-1)N/p\sb 1