Semistablity of syzygy bundles on projective spaces in positive characteristics

Details Semistablity of syzygy bundles on projective spaces in positive characteristics Semistablity of syzygy bundles on projective spaces in positive characteristics

2008-04-03

arxiv.org/abs/0804.0547v2

Mathematics

Algebraic Geometry

25 pages, new version. Also gives estimate on mu-max, and gives a version of Langer's theorem in all degrees and characteristi

Scientific paper

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.

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