The action of the Frobenius map on rank 2 vector bundles over genus 2 curves in small characteristics
Résumé
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.
Domaines
Géométrie algébrique [math.AG]
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