The Morphism Induced by Frobenius Push-Forwards

TL;DR

This paper studies the morphism induced by Frobenius push-forwards on moduli spaces of stable vector bundles over algebraic curves in positive characteristic, showing it is proper and sometimes an embedding, with applications to the structure of certain stable bundles.

Contribution

It establishes that the Frobenius push-forward induces a proper morphism between moduli spaces and a closed immersion under certain conditions, revealing new geometric properties of these spaces.

Findings

The Frobenius push-forward map is a proper morphism between moduli spaces.

For genus at least 2, this map is a closed immersion.

The locus of stable bundles with maximal Harder-Narasimhan polygon is isomorphic to the Jacobian.

Abstract

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