设$X$是定义在特征$p$的代数闭域上的亏格为2的曲线，$X\_1$是它的$p$ -扭.令$M\_X$（分别$M\_{X\_1}$）是$X$上具有平凡行列式的秩为2的半稳定向量丛的（粗）模空间（分别为. $X\_1$）。模空间$M\_X$同构于3维射影空间，并被赋予$X$上的2阶线丛群$J[2]$的作用。当$3\leq p \leq 7$，我们表明，Vernebung（即，Frobenius通过拉回的作用的可分离部分）$V : M\_{X\_1} \dashrightarrow M\_X$完全由其对在$J[2]$的非零元素的作用下不变的线的限制来确定。这些线对应于出现为Prym变种的椭圆曲线，并且Vernebung限制于由$p$相乘引起的态射。因此，我们能够计算的显式方程的Vernebung时，基场具有特征3，5或7。
英文摘要：
Let $X$ be genus 2 curve defined over an algebraically closed field of characteristic $p$ and let $X\_1$ be its $p$-twist. Let $M\_X$ (resp. $M\_{X\_1}$) be the (coarse) moduli space of semi-stable rank 2 vector bundles with trivial determinant over $X$ (resp. $X\_1$). The moduli space $M\_X$ is isomorphic to the 3-dimensional projective space and is endowed with an action of the group $J[2]$ of order 2 line bundles over $X$. When $3\leq p \leq 7$, we show that the Verschiebung (i.e., the separable part of the action of Frobenius by pull-back) $V : M\_{X\_1} \dashrightarrow M\_X$ is completely determined by its restrictions to the lines that are invariant under the action of a non zero element of $J[2]$. Those lines correspond to elliptic curves that appear as Prym varieties and the Verschiebung restricts to the morphism induced by multiplication by $p$. Therefore, we are able to compute the explicit equations of the Verschiebung when the base field has characteristic 3, 5 or 7.
The action of the Frobenius map on rank 2 vector bundles over a supersingular genus 2 curve in characteristic 2
设$X$是特征为2的代数闭域上的光滑真亏格2曲线.绝对Frobenius在$X$上半稳定秩为2的向量丛的模空间$M\_X$上诱导出一个有理映射$F$，它同构于一个三维射影空间. Y. Laszlo和C. Pauly最近给出了普通$X$的$F$方程。利用变形，我们给出了超奇异$X$的这些方程，并得出了一些结论，如$F$的基轨迹（一点），或互补Zaplski开集的稳定性.
Let $X$ be a smooth projective curve of genus $g(X)\geq 1$ over an algebraically closed field $k$ of characteristic $p>0$, $\M^s_X(r,d)$ the moduli space of stable vector bundles of rank $r$ and degree $d$ on $X$. We study the Frobenius stratification of $\M^s_X(r,d)$ in terms of Harder-Narasimhan polygons of Frobenius pull backs of stable vector bundles and obtain the irreducibility and dimension of each non-empty Frobenius stratum in case $(p,g,r)=(3,2,3)$.
Let $X$ be a smooth projective curve of genus $g\geq 2$ over an algebraically closed field $k$ of characteristic $p>0$, $F_X:X\rightarrow X$ the absolute Frobenius morphism. Let $\M^s_X(r,d)$ be the moduli space of stable vector bundles of rank $r$ and degree $d$ on $X$. We study the Frobenius stratification of $\M^s_X(3,0)$ in terms of Harder-Narasimhan polygons of Frobenius pull backs of stable vector bundles and obtain the irreducibility and dimension of each non-empty Frobenius stratum in the case $(p,g)=(3,2)$.
We give explicit formulas for the Kawazumi-Zhang invariant and Faltings delta-invariant of a compact and connected Riemann surface of genus three. The formulas are in terms of two integrals over the associated jacobian, one integral involving the standard Riemann theta function, and another involving a theta function particular to genus three that was discovered by Frobenius. We review part of Frobenius' work on his theta function and connect our results with a formula due to Bloch, Hain and Bost describing the archimedean height pairing of Ceresa cycles in genus three.
In this paper, we study an algebraic fiber space in positive characteristic whose generic fiber $F$ has finitely generated canonical ring and sufficiently large Frobenius stable canonical ring. An example of such a case is when $F$ is $F$-pure and its dualizing sheaf is invertible and ample. We treat a Fujita-type conjecture due to Popa and Schnell concerning direct images of pluricanonical bundles, and prove it under some additional hypotheses. As an application, we show the subadditivity of Kodaira dimensions in some new cases.
We analyze the deformation theory of equivariant vector bundles. In particular, we provide an effective criterion for verifying whether all infinitesimal deformations preserve the equivariant structure. As an application, using rigidity of the Frobenius homomorphism of general linear groups, we prove that projectivizations of Frobenius pullbacks of tautological vector bundles on Grassmanians are arithmetically rigid, that is, do not lift over rings where $p \neq 0$. This gives the same conclusion for Totaro's examples of Fano varieties violating Kodaira vanishing. We also provide an alternativ…
De Concini-Procesi introduced varieties known as wonderful compactifications, which are smooth projective compactifications of semisimple adjoint groups $G$. We study the Frobenius pushforwards of invertible sheaves on the wonderful compactifications, and in particular its decomposition into locally free subsheaves. We give necessary and sufficient conditions for a specific line bundle to be a direct summand of the Frobenius pushforward of another line bundle, formulated in terms of the weight lattice of $\widetilde{G}$, the universal cover of $G$ (identified with the Picard group of the wonde…
Hausel introduced a commutative algebra -- the multiplicity algebra -- associated to a fixed point of the C^*-action on the Higgs bundle moduli space. Here we describe this algebra for a fixed point consisting of a very stable rank 2 vector bundle and zero Higgs field for a curve of low genus. Geometrically, the relations in the algebra are described by a family of quadrics and we focus on the discriminant of this family, providing a new viewpoint on the moduli space of stable bundles. The discriminant in our examples demonstrates that as the bundle varies, we obtain a continuous variation in …
