An analogous of Jouanolou's Theorem in positive characteristic

不变超曲面

Invariant hypersurfaces

该论文证明了一个统一定理：在特征零域上，若两个支配有理映射的无穷多个超曲面逆像一致，则存在非常数有理函数使两者复合相等，并推广了Jouanolou-Hrushovski和Cantat的结果。

Invariant hypersurfaces for derivations in positive characteristic

Let $A$ be an integral $k$-algebra of finite type over an algebraically closed field $k$ of characteristic $p>0$. Given a collection ${\cal{D}}$ of $k$-derivations on $A$, that we interpret as algebraic vector fields on $X=Spec(A)$, we study the group spanned by the hypersurfaces $V(f)$ of $X$ invariant for ${\cal{D}}$ modulo the rational first integrals of ${\cal{D}}$. We prove that this group is always a finite $\mathbb{Z}/p$-vector space, and we give an estimate for its dimension. This is to be related to the results of Jouanolou and others on the number of hypersurfaces invariant for a fol…

正特征下2-generic行列式簇的Nash吹胀

Nash blowups of 2-generic determinantal varieties in positive characteristic

本文通过显式描述toric结构并证明Nash吹胀组合学无特征依赖，得出正特征域上2-generic行列式簇的Nash吹胀非奇异。

Bernstein-Sato theory for arbitrary ideals in positive characteristic

Mustaţă defined Bernstein-Sato polynomials in prime characteristic for principal ideals and proved that the roots of these polynomials are related to the $F$-jumping numbers of the ideal. This approach was later refined by Bitoun. Here we generalize these techniques to develop analogous notions for the case of arbitrary ideals and prove that these have similar connections to $F$-jumping numbers.

On torsors under abelian varieties

Let A be an abelian variety over a local field K of mixed characteristic and with algebraically closed residue field. We provide a geometric construction (via the relative Picard functor) of the Shafarevich duality between the group of isomorphism classes of torsors under A and the "fundamental group" of the Néron model of the dual abelian variety A'. An analogous construction works over fields of positive characteristic p providing a duality on the prime-to-p parts.

Bernstein-Sato polynomials in positive characteristic

In characteristic zero, the Bernstein-Sato polynomial of a hypersurface can be described as the minimal polynomial of the action of an Euler operator on a suitable D-module. We consider the analogous D-module in positive characteristic, and use it to define a sequence of Bernstein-Sato polynomials (corresponding to the fact that we need to consider also divided powers Euler operators). We show that the information contained in these polynomials is equivalent to that given by the F-jumping exponents of the hypersurface, in the sense of Hara and Yoshida.