Beyond a conjecture of Clemens

TL;DR

This paper establishes lower bounds on twists of the canonical bundle for subvarieties in hypersurfaces, proving that generic sextic threefolds lack rational, elliptic, and genus 2 curves, advancing understanding of their geometric properties.

Contribution

It provides new lower bounds on canonical bundle twists and proves the absence of certain curves in generic sextic threefolds, extending previous conjectures.

Findings

Generic sextic threefolds contain no rational or elliptic curves

No nondegenerate curves of genus 2 exist in these threefolds

Lower bounds on canonical bundle twists are established

Abstract

We prove some lower bounds on certain twists of the canonical bundle of a codimension-2 subvariety of a generic hypersurface in projective space. In particular we prove that the generic sextic threefold contains no rational or elliptic curves and no nondegenerate curves of genus 2.