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

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.