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Free curves in Fano hypersurfaces must have high degree

Raymond Cheng

Abstract

The purpose of this note is to show that the minimal $e$ for which every smooth Fano hypersurface of dimension $n$ contains a free rational curve of degree at most $e$ cannot be bounded by a linear function in $n$ when the base field has positive characteristic. This is done by providing a super-linear bound on the minimal possible degree of a free curve in certain Fermat hypersurfaces.

Free curves in Fano hypersurfaces must have high degree

Abstract

The purpose of this note is to show that the minimal for which every smooth Fano hypersurface of dimension contains a free rational curve of degree at most cannot be bounded by a linear function in when the base field has positive characteristic. This is done by providing a super-linear bound on the minimal possible degree of a free curve in certain Fermat hypersurfaces.
Paper Structure (2 sections, 6 theorems, 16 equations)

This paper contains 2 sections, 6 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. Free curves in the Fermat hypersurface

Key Result

Theorem 1

For any algebraically closed field $\mathbf{k}$ of characteristic $p > 0$,

Theorems & Definitions (9)

  • Theorem
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • proof
  • Remark 4
  • Theorem 5
  • proof
  • Theorem 6