The Morphism Induced by Frobenius Push-Forwards

Abstract

Let $X$ be a smooth projective curve of genus $g(X)\ge 1$ over an algebraically closed field $k$ of characteristic $p>0$ and $F_{X/k}:X\to X^{(1)}$ be the relative Frobenius morphism. Let ${\mathfrak{M}}_X^{s(ss)}(r,d)$ (resp. ${\mathfrak{M}}_{X^{(1)}}^{s(ss)}(r\cdot p,d+r(p-1)(g-1))$) be the moduli space of (semi)-stable vector bundles of rank $r$ (resp. $r\cdot p$) and degree $d$ (resp. $d+r(p-1)(g-1)$) on $X$ (resp. $X^{(1)}$). We show that the set-theoretic map $S_{\mathrm{Frob}}^{ss}:{\mathfrak{M}}_X^{ss}(r,d)\to {\mathfrak{M}}_{X^{(1)}}^{ss}(r\cdot p,d+r(p-1)(g-1))$ induced by $[\E]\mapsto[{F_{X/k}}_*(\E)]$ is a proper morphism. Moreover, if $g(X)\ge 2$, the induced morphism $S_{\mathrm{Frob}}^s:{\mathfrak{M}}_X^s(r,d)\to {\mathfrak{M}}_{X^{(1)}}^s(r\cdot p,d+r(p-1)(g-1))$ is a closed immersion. As an application, we obtain that the locus of moduli space ${\mathfrak{M}}_{X^{(1)}}^s(p,d)$ consists of stable vector bundles whose Frobenius pull back have maximal Harder-Narasimhan Polygon is isomorphic to Jacobian variety $\Jac_X$ of $X$.

References in corpus (1)

• Direct images of bundles under Frobenius morphisms