Deformation of Quintic Threefolds to the Chordal Variety

TL;DR

This paper studies the deformation of quintic threefolds to a reducible variety, analyzing stable maps and proving the existence of rigid stable maps of various genera and degrees to general quintics.

Contribution

It extends the analysis of stable morphisms to reducible fibers and establishes the existence of rigid stable maps of arbitrary genus and high degree to very general quintic threefolds.

Findings

Describes the space of genus zero stable morphisms to the central fiber.

Proves the existence of rigid stable maps of arbitrary genus and high degree.

Extends the analysis of stable morphisms to reducible threefolds.

Abstract

We consider a family of quintic threefolds specializing to a certain reducible threefold. We describe the space of genus zero stable morphisms to the central fiber (as defined by J. Li). As an elementary application of an extension of the analysis, we prove the existence of rigid stable maps of arbitrary genus and sufficiently high degree to very general quintics.