Semistablity of syzygy bundles on projective spaces in positive characteristics

TL;DR

This paper proves the semistability of certain syzygy bundles on projective spaces over fields of positive characteristic for infinitely many degrees and provides estimates on their maximal slope, extending Langer's restriction theorem.

Contribution

It establishes the semistability of syzygy bundles for an infinite set of degrees and offers bounds on their maximal slope in arbitrary characteristic, removing previous restrictions.

Findings

Semistability of $ ext{V}_d$ for infinitely many $d$.

Good estimates on $ ext{max}( ext{mu}( ext{V}_d^*))$ in terms of $d$ and $n$.

Extension of Langer's theorem to arbitrary characteristic.

Abstract

In char $k = p >0$, A. Langer proved a strong restriction theorem (in the style of H. Flenner) for semistable sheaves to a very general hypersurface of degree $d$, on certain varieties, with the condition that `char $k > d$'. He remarked that to remove this condition, it is enough to answer either of the following questions affirmatively: {\it For the syzygy bundle $\sV_d$ of ${\mathcal O}(d)$, is $\sV_d$ semistable for arbitrary $n, d$ and $p = {char} k$?, or is there a good estimate on $\mu_{max}(\sV_d^*)$?} Here we prove that (1) the bundle $\sV_d$ is semistable, for a certain infinite set of integers $d\geq 0$, and (2) for arbitrary $d$, there is a good enough estimate on $\mu_{max}(\sV_d^*)$ in terms of $d$ and $n$. In particular one obtains Langer's theorem, in arbitrary characeristic.