Related papers: Elimination with applications to singularities in …
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 discuss the use of field theory for the exact determination of universal properties in two-dimensional statistical mechanics. After a compact derivation of critical exponents of main universality classes, we turn to the off-critical…
We study conformal field theory on two-dimensional orbifolds and show this to be an effective way to analyze physical effects of geometric singularities with angular deficits. They are closely related to boundaries and cross caps.…
The theory of abstract convexity, also known as convexity without linearity, is an extension of the classical convex analysis. There are a number of remarkable results, mostly concerning duality, and some numerical methods, however, this…
The problem of resolution of singularities in positive characteristic can be reformulated as follows: Fix a hypersurface $X$, embedded in a smooth scheme, with points of multiplicity at most $n$. Let an $n$-sequence of transformations of…
I begin by explaining to non-specialists why resolution of singularities in characteristic 0 works. Then I go into some ideas telling how it actually works. I finish with a brief discussion of related results on foliations. I report on work…
We formulate a resolution of singularities algorithm for analyzing the zero sets of real-analytic functions in dimensions $\geq 3$. Rather than using the celebrated result of Hironaka, the algorithm is modeled on a more explicit and…
We develop an extension of valuations theorem for suitable extensions of idempotent semirings. As an application, we give a new proof for the classical case of fields. Along the way, we develop characteristic one analogues of some central…
A new proof of equivariant resolution of singularities under a finite group action in characteristic 0 is provided. We assume we know how to resolve singularities without group action. We first prove equivariant resolution of toroidal…
We build on our previous paper \cite{constructive} by using the general method introduced there in conjunction with invariant theory. This yields quantifier elimination results for the classical quaternions, octonions, as well as other…
We present various results on disconnected reductive groups, in particular about the characteristic 0 representation theory of such groups over finite fields.
We discuss the birational geometry of singular surfaces in positive characteristic. More precisely, we establish the minimal model program and the abundance theorem for Q-factorial surfaces and for log canonical surfaces. Moreover, in the…
We study existential theories of henselian valued fields of positive characteristic with parameters from a trivially valued subfield. Compared to previous work, we relax perfectness and separability assumptions, and instead work with the…
We answer a question of Abhyankar by constructing an algebraic closure of the field of power series over a field of positive characteristic, using "generalized power series". (The corresponding construction in characteristic 0 dates back to…
We provide axiomatization and relative quantifier elimination for valued fields equipped with an automorphism, in residue characteristic zero. Similar results are known under strong assumptions on the interaction between the automorphism…
We present a simple proof of the well-known fact concerning the number of solutions of diagonal equations over finite fields. In a similar manner, we give an alternative proof of the recent result on generalizations of Carlitz equations. In…
This article has the following aims: (1) Extend the notion of fuchsian singularities (of first kind) to base fields of arbitrary characteristic. (2) Discuss their relationship to mathematical objects of a different nature. (3) Provide a…
In this paper we study the equations of the elimination ideal associated with $n+1$ generic multihomogeneous polynomials defined over a product of projective spaces of dimension $n$. We first prove a duality property and then make this…
We study the negative $K$-theory of singular varieties over a field of positive characteristic and in particular, prove the vanishing of $K_i(X)$ for $i < -d-2$ for a $k$-variety of dimension $d$.
In characteristic zero, we construct logarithmic resolution of singularities, with simple normal crossings exceptional divisor, using weighted blow-ups.