Strongly residual coordinates over A[x]

For a domain A of characteristic zero, a polynomial f over A[x] is called a strongly residual coordinate if f becomes a coordinate (over A) upon going modulo x, and f becomes a coordinate upon inverting x. We study the question of when a strongly residual coordinate is a coordinate, a question closely related to the Dolgachev-Weisfeiler conjecture. It is known that all strongly residual coordinates are coordinates for n=2 . We show that a large class of strongly residual coordinates that are generated by elementaries upon inverting x are in fact coordinates for arbitrary n, with a stronger result in the n=3 case. As an application, we show that all Venereau-type polynomials are 1-stable coordinates.Comment: 15 pages. Some minor clarifications and notational improvements from the first versio

Log canonical thresholds of Del Pezzo Surfaces in characteristic p

The global log canonical threshold of each non-singular complex del Pezzo surface was computed by Cheltsov. The proof used Koll\'ar-Shokurov's connectedness principle and other results relying on vanishing theorems of Kodaira type, not known to be true in finite characteristic. We compute the global log canonical threshold of non-singular del Pezzo surfaces over an algebraically closed field. We give algebraic proofs of results previously known only in characteristic $0$. Instead of using of the connectedness principle we introduce a new technique based on a classification of curves of low degree. As an application we conclude that non-singular del Pezzo surfaces in finite characteristic of degree lower or equal than $4$ are K-semistable.Comment: 21 pages. Thorough rewrite following referee's suggestions. To be published in Manuscripta Mathematic

Equivalent birational embeddings II: divisors

Two divisors in $\P^n$ are said to be Cremona equivalent if there is a Cremona modification sending one to the other. We produce infinitely many non equivalent divisorial embeddings of any variety of dimension at most 14. Then we study the special case of plane curves and rational hypersurfaces. For the latter we characterise surfaces Cremona equivalent to a plane.Comment: v2 Exposition improved, thanks to referee, unconditional characterization of surfaces Cremona equivalent to a plan

On Weierstra{\ss} semigroups at one and two points and their corresponding Poincar\'e series

The aim of this paper is to introduce and investigate the Poincar\'e series associated with the Weierstra{\ss} semigroup of one and two rational points at a (not necessarily irreducible) non-singular projective algebraic curve defined over a finite field, as well as to describe their functional equations in the case of an affine complete intersection.Comment: Beginning of Section 3 and Subsection 3.1 were modifie