arXiv subjects
Publications and source records attributed to Georg Hein.
We describe the possible restrictions of the cotangent bundle Ω_{\PP^N} to an elliptic curve C \subset \PP^N. We apply this in positive characteristic to the computation of the Hilbert-Kunz function of a homogeneous R_+-primary ideal I \subset R in the graded section ring R = \bigoplus_{n \in \NN} Γ(C, Ø(n)).
We develop a formula (Theorem 5.1) which allows to compute top Chern classes of vector bundles on the vanishing locus $V(s)$ of a section of this bundle. This formula particularly applies in the case when $V(s)$ is the union of locally complete intersections giving the individual contribution of each component and their mutual intersections. We conclude with applications to the enumeration of rational curves in complete intersections in projective space.
Let $X$ be a smooth variety defined over an algebraically closed field of arbitrary characteristic and $Ø_X(H)$ be a very ample line bundle on $X$. We show that for a semistable $X$-bundle $E$ of rank two, there exists an integer $m$ depending only on $Δ(E).H^{\dim(X)-2}$ and $H^{\dim(X)}$ such that the restriction of $E$ to a general divisor in $|mH|$ is again semistable. As corollaries we obtain boundedness results, and weak versions of Bogomolov's theorem and Kodaira's vanishing theorem for surfaces in arbitrary characteristic.
We show for the moduli space of rank-2 coherent sheaves on an algebraic surface that there exists a 'dual' moduli space. This dual space allows a construction of the first one without using the GIT construction. Furthermore, we obtain a Barth-morphism, generalizing the concept of jumping lines. This morphism is by construction a finite morphism.