The morphism induced by Frobenius push-forward |
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Authors: | LingGuang Li |
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Institution: | 1. Department of Mathematics, Tongji University, Shanghai, 200092, China 2. School of Mathematical Sciences, Fudan University, Shanghai, 200433, China
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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. |
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Keywords: | Frobenius morphism stable vector bundle moduli space stratification |
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