Related papers: Moduli theory and arithmetic of algebraic varietie…

In this letter we present some new results on modular theory and its application in quantum field theory. In doing this we develop some new proposals how to generalize concepts of geometrical action. Therefore the spirit of this letter is…

Working in positive characteristic, we show how one can use information about the dimension of moduli spaces of rational curves on a Fano variety $X$ over $\mathbb{F}_q$ to obtain strong estimates for the number of $\mathbb{F}_q(t)$-points…

We establish sharp estimates that adapt the polynomial method to arbitrary varieties. These include a partitioning theorem, estimates on polynomials vanishing on fixed sets and bounds for the number of connected components of real algebraic…

We study equivariant geometry and rationality of moduli spaces of points on the projective line, for twists associated with permutations of the points.

The double point relation defines a natural theory of algebraic cobordism for bundles on varieties. We construct a simple basis (over the rationals) of the corresponding cobordism groups over Spec(C) for all dimensions of varieties and…

It is proved that the number of deformation types of complex structures on a fixed oriented smooth four-manifold can be arbitrarily large. The considered examples are locally simple abelian covers of rational surfaces.

We study the distribution of integral points on log varieties.

By a heuristic argument, we relate two conjectures. One is a version of Manin's conjecture about the distribution of rational points on a Fano variety. We concern specific singular Fano varieties, namely quotients of projective spaces by…

This is an expository article on the theory of formal group laws in homotopy theory, with the goal of leading to the connection with higher-dimensional abelian varieties and automorphic forms. These are roughly based on a talk at the…

This is mostly* a non-technical exposition of the joint work arXiv:1212.0373 with Caporaso and Payne. Topics include: Moduli of Riemann surfaces / algebraic curves; Deligne-Mumford compactification; Dual graphs and the combinatorics of the…

Monads are of interest both in semantics and in higher dimensional algebra. It turns out that the idea behind usual notion finitary monads (whose values on all sets can be computed from their values on finite sets) extends to a more general…

A simple formula computing the multiplier ideal of a monomial ideal on an arbitrary affine toric variety is given. Variants for the multiplier module and test ideals are also treated.

We define a monoidal semantics for algebraic theories. The basis for the definition is provided by the analysis of the structural rules in the term calculus of algebraic languages. Models are described both explicitly, in a form that…

The circle method has been successfully used over the last century to study rational points on hypersurfaces. More recently, a version of the method over function fields, combined with spreading out techniques, has led to a range of results…

The moduli spaces refered to are topological spaces whose path components parametrize homotopy types. Such objects have been studied in two separate contexts: rational homotopy types, in the work of several authors in the late 1970's; and…

It is shown that rational points over finite fields of moduli spaces of stable quiver representations are counted by polynomials with integer coefficients. These polynomials are constructed recursively using an identity in the Hall algebra…

This survey paper was primarily written as as the support for a course pesented at the JNCF2025: it aims to present some material that illustrates the kind of estimates one can obtain in effective algebraic geometry, for affine polynomial…

The goal of this paper is to consider some relations between varieties of representations of groups and varieties of associative algebras. The main emphasis is put on the varieties of representations of groups induced by the varieties of…

In this note, we give a new simple system of global parameters on the moduli space of rational functions, and clarify the relation to the parameters indicating location of fixed points and the indices at them. As a byproduct, we solve a…

We investigate commutator operations on compatible uniformities of an algebra. We present a commutator operation for compatible uniformities of an algebra in a congruence-modular variety which extends the commutator on congruences, and…