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 foliation of codimension 1. As an application, given a $k$-algebra $B$ between $A^p$ and $A$, we show that the kernel of the pull-back morphism $Pic(B)\rightarrow Pic(A)$ is a finite $\mathbb{Z}/p$-vector space. In particular, if $A$ is a UFD, then the Picard group of $B$ is finite.Comment: 16 page

A Pontryagin Maximum Principle in Wasserstein Spaces for Constrained Optimal Control Problems

In this paper, we prove a Pontryagin Maximum Principle for constrained optimal control problems in the Wasserstein space of probability measures. The dynamics, is described by a transport equation with non-local velocities and is subject to end-point and running state constraints. Building on our previous work, we combine the classical method of needle-variations from geometric control theory and the metric differential structure of the Wasserstein spaces to obtain a maximum principle stated in the so-called Gamkrelidze form.Comment: 35 page

Cohomology of regular differential forms for affine curves

Let $C$ be a complex affine reduced curve, and denote by $H^1(C)$ its first truncated cohomology group, i.e. the quotient of all regular differential 1-forms by exact 1-forms. First we introduce a nonnegative invariant $\mu'(C,x)$ that measures the complexity of the singularity of $C$ at the point $x$. Then, if $H_1(C)$ denotes the first singular homology group of $C$ with complex coefficients, we establish the following formula: $$dim H^1(C)=dim H_1(C) + \sum_{x\in C} \mu'(C,x)$$ Second we consider a family of curves given by the fibres of a dominant morphism $f:X\to \mathbb{C}$, where $X$ is an irreducible complex affine surface. We analyze the behaviour of the function $y\mapsto dim H^1(f^{-1}(y))$. More precisely, we show that it is constant on a Zariski open set, and that it is lower semi-continuous in general.Comment: 16 page