Toward Clemens' Conjecture in degrees between 10 and 24

TL;DR

This paper proposes a new condition that supports Clemens' conjecture for degrees 10 to 24, suggesting finiteness and specific properties of rational curves on general quintic threefolds.

Contribution

It introduces a likely condition that implies a form of Clemens' conjecture for certain degrees, advancing understanding of rational curves on quintic threefolds.

Findings

Hilbert scheme of rational curves is finite, nonempty, and reduced for degrees 10 to 24.

Each curve has a balanced normal sheaf and maximal rank embedding.

Supports Clemens' conjecture in specified degrees.

Abstract

We introduce and study a likely condition that implies the following form of Clemens' conjecture in degrees $d$ between 10 and 24: given a general quintic threefold $F$ in complex $\IP^4$, the Hilbert scheme of rational, smooth and irreducible curves $C$ of degree $d$ on $F$ is finite, nonempty, and reduced; moreover, each $C$ is embedded in $F$ with balanced normal sheaf $\O(-1)\oplus\O(-1)$, and in $\IP^4$ with maximal rank.