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Extensions and paracanonical curves of genus 6

Frank-Olaf Schreyer, Hoang Le Truong

Abstract

In this note we give a computationally easy to use method to compute a maximal extension of certain varieties. As a application we prove that a general paracanonical curve C genus 6 as a codimension three subvarieties of P^4 extend to precisely 26 families of surfaces Y in P^5.

Extensions and paracanonical curves of genus 6

Abstract

In this note we give a computationally easy to use method to compute a maximal extension of certain varieties. As a application we prove that a general paracanonical curve C genus 6 as a codimension three subvarieties of P^4 extend to precisely 26 families of surfaces Y in P^5.
Paper Structure (6 sections, 12 theorems, 18 equations, 1 table)

This paper contains 6 sections, 12 theorems, 18 equations, 1 table.

Table of Contents

  1. Introduction
  2. Extensions
  3. Extensions of general paracanonical curves of genus $6$
  4. Maximal extensions
  5. Families of pairs $(Y,C)$
  6. Special curves and surfaces

Key Result

Theorem 2.1

Let $X \subset {\Bbb P}^{n}$ be a variety of codimension $c$ which is not a cone, whose minimal free resolution $F$\xymatrix{ 0 & \ar[l] S_{X} &\ar[l] F_{0} & \ar[l] \ldots & \ar[l] F_{i-1} & \ar[l]_{\varphi_{i}} F_{i} &\ar[l]_{\varphi_{i+1}} F_{i+1} & \ar[l] \ldots &\ar[l] F_{pd}& \ar[l] 0 \\ }has for the entries of the $\mathop{\mathrm{rank}}\nolimits F_{i-1}\times \mathop{\mathrm{rank}}\nolimi

Theorems & Definitions (23)

  • Theorem 2.1
  • proof
  • Example 2.2: Pinkham's example Pinkham
  • Example 2.3: Tom and Jerry, Reid
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • ...and 13 more