The morphism induced by Frobenius push-forward

Abstract

Let X be a smooth projective curve of genus g ⩾ 2 over an algebraically closed field k of characteristic p > 0, and F: X → X (1) the relative Frobenius morphism. Let \(\mathfrak{M}_X^s (r,d)\) (resp. \(\mathfrak{M}_X^{ss} (r,d)\)) be the moduli space of (resp. semi-)stable vector bundles of rank r and degree d on X. We show that the set-theoretic map \(S_{Frob}^{ss} :\mathfrak{M}_X^{ss} (r,d) \to \mathfrak{M}_{X^{(1)} }^{ss} (rp,d + r(p - 1)(g - 1))\) induced by Open image in new window is a proper morphism. Moreover, the induced morphism \(S_{Frob}^s :\mathfrak{M}_X^s (r,d) \to \mathfrak{M}_{X^{(1)} }^s (rp,d + r(p - 1)(g - 1))\) is a closed immersion. As an application, we obtain that the locus of moduli space \(\mathfrak{M}_{X^{(1)} }^s (p,d)\) consisting of stable vector bundles whose Frobenius pull backs have maximal Harder-Narasimhan polygons is isomorphic to the Jacobian variety JacX of X.