Laudal's Lemma in positive characteristic

TL;DR

This paper investigates the failure of Laudal's Lemma in positive characteristic fields and establishes new bounds for when a curve lies on a surface of a given degree.

Contribution

It extends Laudal's Lemma to positive characteristic by providing alternative bounds for the degree of curves ensuring they lie on a surface of a specified degree.

Findings

Laudal's Lemma does not hold in positive characteristic.

New bounds d > f(s) are established for positive characteristic.

The results specify conditions under which curves are contained in surfaces of degree s.

Abstract

Laudal's Lemma states that if $C$ is a curve of degree $d > s^2 + 1$ in $\mathbb P^3$ over an algebraically closed field of characteristic 0 such that its plane section is contained in an irreducible curve of degree s, then $C$ lies on a surface of degree $s$. We show that the same result does not hold in positive characteristic and we find different bounds $d > f(s)$ which ensure that $C$ is contained in a surface of degree $s$.