The morphism induced by Frobenius push-forward  

作　　者：LI LingGuang 

机构地区：[1]Department of Mathematics,Tongji University [2]School of Mathematical Sciences,Fudan University

出　　处：《Science China Mathematics》2014年第1期61-67,共7页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China (Grant No. 11271275)

摘　　要：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 M s X(r,d)(resp.M ss X(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 ss Frob:M ss X(r,d)→M ss X(1)(rp,d+r(p-1)(g-1))induced by[E]→[F(E)]is a proper morphism.Moreover,the induced morphism S s Frob:M s X(r,d)→M s X(1)(rp,d+r(p-1)(g-1))is a closed immersion.As an application,we obtain that the locus of moduli space M s X(1)(p,d)consisting of stable vector bundles whose Frobenius pull backs have maximal Harder-Narasimhan polygons is isomorphic to the Jacobian variety Jac X of X.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 M s X（r,d）（resp.M ss X（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 ss Frob：M ss X（r,d）→M ss X（1）（rp,d＋r（p-1）（g-1））induced by[E]→[F（E）]is a proper morphism.Moreover,the induced morphism S s Frob：M s X（r,d）→M s X（1）（rp,d＋r（p-1）（g-1））is a closed immersion.As an application,we obtain that the locus of moduli space M s X（1）（p,d）consisting of stable vector bundles whose Frobenius pull backs have maximal Harder-Narasimhan polygons is isomorphic to the Jacobian variety Jac X of X.

关 键 词：Frobenius morphism stable vector bundle moduli space STRATIFICATION 14H60 14D20 13A35 

分 类 号：O186.11[理学—数学]