arXiv subjects
Publications and source records attributed to Laurent Ducrohet.
We study stable rank 2 vector bundles with trivial determinant whose Frobenius pull back is non stable over a general curve of genus g>1. In genus 2, we apply recent results about the theta divisor associated to the bundle B of locally exact differential forms and we derive a scheme-theoretic description of the Frobenius inverse image of the semi-stable boundary of the moduli space.
Let X be a general proper and smooth curve of genus 2 (resp. of genus 3) defined over an algebraically closed field of characteristic p. When 3\leq p \leq 7, the action of Frobenius on rank 2 semi-stable vector bundles with trivial determinant is completely determined by its restrictions to the 30 lines (resp. the 126 Kummer surfaces) that are invariant under the action of some order 2 line bundle over X. Those lines (resp. those Kummer surfaces) are closely related to the elliptic curves (resp. the abelian varieties of dimension 2) that appear as the Prym varieties associated to double étale coverings of X. We are therefore able to compute explicit equations of this action in these cases. We perform some of these computations and draw some consequences.
Let $X$ be genus 2 curve defined over an algebraically closed field of characteristic $p$ and let $X\_1$ be its $p$-twist. Let $M\_X$ (resp. $M\_{X\_1}$) be the (coarse) moduli space of semi-stable rank 2 vector bundles with trivial determinant over $X$ (resp. $X\_1$). The moduli space $M\_X$ is isomorphic to the 3-dimensional projective space and is endowed with an action of the group $J[2]$ of order 2 line bundles over $X$. When $3\leq p \leq 7$, we show that the Verschiebung (i.e., the separable part of the action of Frobenius by pull-back) $V : M\_{X\_1} \dashrightarrow M\_X$ is completely determined by its restrictions to the lines that are invariant under the action of a non zero element of $J[2]$. Those lines correspond to elliptic curves that appear as Prym varieties and the Verschiebung restricts to the morphism induced by multiplication by $p$. Therefore, we are able to compute the explicit equations of the Verschiebung when the base field has characteristic 3, 5 or 7.
Let $X$ be a smooth proper genus 2 curve over an algebraically closed field of characteristic 2. The absolute Frobenius induces a rational map $F$ on the the moduli space $M\_X$ of semi-stable rank 2 vector bundles over $X$, which is isomorphic to a 3-dimensional projective space. Y. Laszlo and C. Pauly recently gave the equations of $F$ for an ordinary $X$. Using deformation, we give these equations for a supersingular $X$ and draw some consequences such as the base locus of $F$ (one point), or the stability of the complementary Zariski open set.