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 $X$.
Categories: math.AG