An infinitesimal approach to a conjecture of Eisenbud and Harris

TL;DR

This paper explores a deformation approach to Eisenbud and Harris's conjecture, showing that certain algebraic curves deform along with point sets under specific conditions, advancing understanding of the conjecture's geometric implications.

Contribution

It introduces an infinitesimal deformation method to analyze the conjecture, linking point conditions on quadrics to the deformation behavior of algebraic curves.

Findings

Curve deforms with point set under constant quadrics conditions

Deformation preserves the hyperplane section's low degree property

Provides new insights into the geometric structure of high genus curves

Abstract

Eisenbud and Harris conjectured in 1982 that an algebraic curve of high genus lies on a surface of low degree (which they proved for curves of very large degree). They observed constraints on the Hilbert function of a general hyperplane section which imply that the hyperplane section lies on a curve of low degree. We investigate this situation under deformation. Given a set of sufficiently many points (as postulated by the conjecture) on a linearly normal curve, we show that if the number of conditions on quadrics remains constant, then for every deformation of the points the curve deforms along with them.