Curves with stable or semistable normal bundle on Fano hypersurfaces
Ziv Ran
TL;DR
This work addresses the problem of curves with (semi)stable normal bundles on hypersurfaces by developing a degeneration framework built around fish/fang configurations and parallel transport. It proves that for $n\ge3$ and genus $g\ge1$, there exist genus-$g$ curves of large degree $e$ in general hypersurfaces of degree $n$ in $\mathbb{P}^n$ (and in $\mathbb{P}^n$ itself) whose normal bundle $N_{C/X}$ is stable, with sufficiently general full-rank subsheaves also stable; when $g=1$, $N$ is semistable. For hypersurfaces of degree $d< n$, a congruence/divisibility condition yields an arithmetic progression of degrees $e$ for which such semistable curves exist. The approach combines inductive degeneration arguments, openness of stability, and careful analysis of modifications and Harder–Narasimhan filtrations, yielding new non-integer-slope semistability results beyond the classical integral-slope cases. The results broaden the landscape of known (semi)stable normal bundles on hypersurfaces, with particular impact on understanding normal bundles in Fano and anticanonical settings and providing tools potentially useful for Brill–Noether-type questions in higher dimensions.
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 $¶^n$