Related papers: Strong resolution of singularities in characterist…

Let $X$ be a closed subscheme embedded in a scheme $W$ smooth over a field ${\bf k}$ of characteristic zero, and let ${\mathcal I}(X)$ be the sheaf of ideals defining $X$. Assume that the set of regular points of $X$ is dense in $X$. We…

We introduce a new fundamental group scheme for varieties defined over an algebraically closed field of positive characteristic and we use it to study generalization of some of C. Simpson's results to positive characteristic. We also study…

For a totally positive definite quadratic form over the ring of integers of a totally real number field $K$, we show that there are only finitely many totally real field extensions of $K$ of a fixed degree over which the form is universal…

This work is devoted to prove uniqueness result for the positive solution to a strongly competing system of Lotka - Volterra type in the limiting configuration, when the competition rate tends to infinity.

In this survey paper we give an overview on some aspects of singularities of algebraic varieties over an algebraically closed field of arbitrary characteristic. We review in particular results on equisingularity of plane curve…

We produce a class of $\omega$-categorical structures with finite signature by applying a model-theoretic construction -- a refinement of the Hrushosvki-encoding -- to $\omega$-categorical structures in a possibly infinite signature. We…

A smooth scheme X over a field k of positive characteristic is said to be strongly liftable, if X and all prime divisors on X can be lifted simultaneously over W_2(k). In this paper, we give some concrete examples and properties of strongly…

This article contains an elementary constructive proof of resolution of singularities in characteristic zero. Our proof applies in particular to schemes of finite type and to analytic spaces (so we recover the great theorems of Hironaka).…

The Bernstein-Sato polynomial is an important invariant of an element or an ideal in a polynomial ring or power series ring of characteristic zero, with interesting connections to various algebraic and topological aspects of the…

Let $X$ be an algebraic variety with Gorenstein singularities. We define the notion of a wonderful resolution of singularities of $X$ by analogy with the theory of wonderful compactifications of semi-simple linear algebraic groups. We prove…

We prove an embedded local uniformization theroem for a valuation centered on a point of a quasi-excellent scheme of characteristic zero. The proof reduces to valuations of rank 1 and consists in desingularizing the ideal formed by the…

Using a version of Hironaka's resolution of singularities for real-analytic functions, any elliptic multiplier $\mathrm{Op}(p)$ of order $d>0$, real-analytic near $p^{-1}(0)$, has a fundamental solution $\mu_0$. We give an integral…

We show how singularities shape the evolution of rational discrete dynamical systems. The stabilisation of the form of the iterates suggests a description providing among other things generalised Hirota form, exact evaluation of the…

For a normal F-finite variety $X$ and a boundary divisor $\Delta$ we give a uniform description of an ideal which in characteristic zero yields the multiplier ideal, and in positive characteristic the test ideal of the pair $(X,\Delta)$.…

We prove a unique continuation result for an ill-posed characteristic problem. A model problem of this type occurs in A.D.~Ionescu \& S.~Klainerman article (Theorem 1.1 in \cite{MR2470908}) and we extend their model-result using only…

In this paper, we study the set of positive integers that characterize the universality of $m$-gonal form.

In this article we extend independent results of Lusztig and H\'ezard concerning the existence of irreducible characters of finite reductive groups, (defined in good characteristic and arising from simple algebraic groups), satisfying a…

We prove that the Hilbert scheme of the plane in positive characteristic admits an invertible top differential form. This implies certain integrability properties of the symmetric powers of the plane. This allows to define a function on the…

Let $K$ be a Birch field, that is, a field for which every diagonal form of odd degree in sufficiently many variables admits a non-zero solution; for example, $K$ could be the field of rational numbers. Let $f_1, \ldots, f_r$ be homogeneous…

We give an overview of the fundamental definitions and results concerning hypersurface singularities, defined by convergent power series over an arbitrary real valued field. This approach combines, on the one hand, the classical case of…