Curves with stable or semistable normal bundle on Fano hypersurfaces

TL;DR

This paper proves the existence of curves with stable or semistable normal bundles on general Fano hypersurfaces, extending previous results to broader cases and providing explicit conditions on degrees and genera.

Contribution

It establishes the existence of such curves with stable or semistable normal bundles on general Fano hypersurfaces, generalizing prior work limited to ambient projective space.

Findings

Existence of curves with stable normal bundles on Fano hypersurfaces for large degrees.

Construction of curves with semistable normal bundles for genus 1.

Identification of arithmetical conditions on degree and genus for such curves.

Abstract

For every $n\geq 3, g\geq 1$ and all large enough $e$ depending on $n,g$, there exist curves of genus $g$, degree $e$ in a general hypersurface of degree $n$ in $\mathbb P^n$, or in $\mathbb P^n$ itself, whose whose normal bundle $N$ is stable, as is any sufficiently general full-rank subsheaf of $N$. For $g=1$, $N$ is semi-stable. On general hypersurface of degree $d< n$ in $\mathbb P^n$, such that a certain arithmetical condition on $d,n, g $ holds, there exists an arithmetical progression of $e$ values so that curves of degree $e$ and genus $g$ with semistable normal bundle exist. Previous results were restricted to certain cases with ambient space $\P^n$