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A family of Non-Weierstrass Semigroups

David Eisenbud, Frank-Olaf Schreyer

Abstract

A numerical semigroup is said to be Weierstrass if it is the semigroup of pole orders of rational functions that are regular at all but one point of some compact Riemann surface or smooth algebraic curve. Hurwitz asked in 1892 whether all numerical semigroups can occur. In this paper we give a new method, using syzygies,to show that certain semigroups are not Weierstrass, including the first one of multiplicity 6 (the lowest possible) and genus 13 (the lowest known). We give many other examples to which the method applies.

A family of Non-Weierstrass Semigroups

Abstract

A numerical semigroup is said to be Weierstrass if it is the semigroup of pole orders of rational functions that are regular at all but one point of some compact Riemann surface or smooth algebraic curve. Hurwitz asked in 1892 whether all numerical semigroups can occur. In this paper we give a new method, using syzygies,to show that certain semigroups are not Weierstrass, including the first one of multiplicity 6 (the lowest possible) and genus 13 (the lowest known). We give many other examples to which the method applies.
Paper Structure (5 sections, 7 theorems, 16 equations)

This paper contains 5 sections, 7 theorems, 16 equations.

Table of Contents

  1. Pinkham's theorem and a structure theorem
  2. Pinkham's Theorem
  3. A structure theorem for resolutions
  4. Special resolutions and Non-Weierstass Semigroups
  5. Examples

Key Result

Proposition 1.1

Suppose that $C$ is a smooth curve defined over a field $k$, and that $p\in C$ is a $k$-rational point with Weierstrass semigroup $S$. Let $A$ be the coordinate ring of the affine curve $C\setminus \{p\}$. There is a flat family of affine rings $A_{z}$ over $k[z]$ with generic fiber $A\otimes_{k}k(z

Theorems & Definitions (25)

  • Proposition 1.1
  • proof
  • Proposition 1.2
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof : Proof of Proposition \ref{['stability of conditions']}
  • Conjecture 2.4
  • Example 2.5
  • Proposition 2.6
  • ...and 15 more