Frobenius Pullbacks Research Articles

Moduli of Vector Bundles on Curves in Positive Characteristics

Let X be a projective curve of genus 2 over an algebraically closed field of characteristic 2. The Frobenius map on X induces a rational map on the moduli scheme of rank-2 bundles. We show that up to isomorphism, there is only one (up to tensoring by an order two line bundle) semi-stable vector bundle of rank 2 (with determinant equal to a theta characteristic) whose Frobenius pull-back is not semi-stable. The indeterminacy of the Frobenius map at this point can be resolved by introducing Higgs bundles.

Groups of Russell type over a curve

We consider groups of Russell type over a curve, which are forms of G a -bundle over a curve. In this article, we show that groups of Russell type over a curve, not necessarily a Tango curve, are associated with locally free sheaves of rank two on the base curve. Moreover, these locally free sheaves are often stable but their Frobenius pull-backs are not stable. We observe also pathological phenomena on completions of groups of Russell type, namely, the existence of non-closed global differential forms and the non-reducedness of the automorphism group scheme.

Stability of Frobenius pull-backs of tangent bundles and generic injectivity of Gauss maps in positive characteristic

HTML view is not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button. For a smooth projective variety $X$ of dimension $n$ in a projective space ${\bb P}^N$ defined over an algebraically closed field $k$, the Gauss map is a morphism from $X$ to the Grassmannian of $n$-plans in ${\bb P}^N$sending $x \in X$ to the embedded tangent space $T_xX \subset {\bb P}^N$. The purpose of this paper is to prove the generic injectivity of Gauss maps in positive characteristic for two cases; (1) weighted complete intersections of dimension $n \geqslant 3$ of general type; (2) surfaces or 3-folds with $\mu$-semistable tangent bundles; based on a criterion of Kaji by looking at the stability of Frobenius pull-backs of their tangent bundles. The first result implies that a conjecture of Kleiman-Piene is true in case $X$ is of general type of dimension $n \geqslant 3$. The second result is a generalization of the injectivity for curves.

Frobenius pull-back of vector bundles of rank 2 over non-uniruled varieties

Frobenius pull-back of vector bundles of rank 2 over non-uniruled varieties