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Local Complete Intersections and Weierstrass Points

André Contiero, Sarah Mazzini

Abstract

This work presents a simple proof that the moduli space of complete integral Gorenstein curves with a prescribed symmetric Weierstrass semigroup becomes a weighted projective space, even for fields of positive characteristic, when the associated monomial curve is a local complete intersection.

Local Complete Intersections and Weierstrass Points

Abstract

This work presents a simple proof that the moduli space of complete integral Gorenstein curves with a prescribed symmetric Weierstrass semigroup becomes a weighted projective space, even for fields of positive characteristic, when the associated monomial curve is a local complete intersection.
Paper Structure (11 sections, 7 theorems, 64 equations)

This paper contains 11 sections, 7 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. $\mathcal{M}_{g,1}^{\mathrm{S}}$ coming from versal deformation
  3. Reviewing Weierstrass points
  4. Gorenstein curves and subcanonical points
  5. On $P$-hermitian bases
  6. The ideal of the canonically embedded $\mathcal{C}$
  7. Unfolding the defining equations
  8. A variant of Hauser's algorithm
  9. The Main Theorem
  10. Proof of the Main Theorem
  11. Examples

Key Result

Corollary 1.1

A complete intersection numerical semigroup is realized as a Weierstrass semigroup of a smooth curve.

Theorems & Definitions (13)

  • proof
  • Corollary 1.1: Schlessinger SchPhD and Pinkham Pi74
  • Corollary 1.2
  • Remark 3.1
  • Theorem 3.2: Pi74
  • Lemma 3.3: Syzygy Lemma CS
  • Theorem 3.4: c.f. Thm 2.6 of CS
  • Theorem 3.5: c.f. Thm. 2.7 of CS
  • Theorem 4.1
  • proof
  • ...and 3 more