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A characterization of bielliptic curves via syzygy schemes

Marian Aprodu, Andrea Bruno, Edoardo Sernesi

Abstract

We prove that a canonical curve $C$ of genus $\ge 11$ is bielliptic if and only if its second syzygy scheme $\mathrm{Syz}_2(C)$ is different from $C$.

A characterization of bielliptic curves via syzygy schemes

Abstract

We prove that a canonical curve of genus is bielliptic if and only if its second syzygy scheme is different from .

Paper Structure

This paper contains 6 sections, 9 theorems, 60 equations.

Table of Contents

  1. The second syzygy scheme
  2. Canonical curves of genus 6
  3. Plane sextics
  4. Canonical curves either lying on a Del Pezzo surface or bielliptic
  5. Curves in 3-dimensional scrolls
  6. The second syzygy scheme of a 4-gonal canonical curve

Key Result

Theorem 1

Let $C\subset \mathbb P^{g-1}$ be a nonhyperelliptic canonical curve of genus $\ge 11$ and gonality at least $4$. Then $\mathrm{Syz}_2(C)\ne C$ if and only if $C$ is not bielliptic. In the bielliptic case, $\mathrm{Syz}_2(C)$ is an elliptic cone.

Theorems & Definitions (16)

  • Theorem 1
  • Proposition 2
  • proof
  • Lemma 3
  • proof
  • Proposition 4
  • Lemma 5
  • proof
  • Proposition 6
  • proof
  • ...and 6 more