Related papers: Birational geometry of Deligne-Mumford stacks
The aim of this article is to develop the theory of motivic integration over Deligne-Mumford stacks and to apply it to the birational geometry of stacks.
We consider the question of how geometric structures of a Deligne-Mumford stack affect its Gromov-Witten invariants. The two geometric structures studied here are {\em gerbes} and {\em root constructions}. In both cases, we explain…
We discuss invariants in equivariant birational geometry.
We introduce new invariants in equivariant birational geometry and study their relation to modular symbols and cohomology of arithmetic groups.
We construct new invariants of equivariant birational isomorphisms taking values in equivariant Burnside groups.
We investigate equivariant birational geometry of rational surfaces and threefolds from the perspective of derived categories.
Here we investigate the birational geometry of projective varieties of arbitrary dimension having defective higher secant varieties. We apply the classical tool of tangential projections and we determine natural conditions for uniruledness,…
We discuss the equivariant Burnside group and related new invariants in equivariant birational geometry, with a special emphasis on applications in low dimensions.
In this paper we will survey some recent developments in the last decade or so on variation of Geometric Invariant Theory and its applications to Birational Geometry such as the weak Factorization Theorems of nonsingular projective…
We introduce the notion of strong regular embeddings of Deligne-Mumford stacks. These morphisms naturally arise in the related contexts of generalized Euler sequences and hypertoric geometry.
We study dualizing complexes on algebraic stacks. In particular, we show their existence for (tame) Deligne--Mumford stacks of equicharacteristic in great generality.
The Gromov-Witten theory of Deligne-Mumford stacks is a recent development, and hardly any computations have been done beyond 3-point genus 0 invariants. This paper provides explicit recursions which, together with some invariants computed…
We study equivariant birationality from the perspective of derived categories. We produce examples of nonlinearizable but stably linearizable actions of finite groups on smooth cubic fourfolds.
In this expository article we give a categorical definition of the integral cohomology ring of a stack. We show that for quotient stacks the categorical cohomology may be identified with equivariant cohomology. Via this identification we…
We present a construction of Donaldson-Thomas invariants for three-dimensional projective Calabi-Yau Deligne-Mumford stacks. We also study the structure of these invariants for etale gerbes over such stacks.
We give a survey on higher invariants in noncommutative geometry and their applications to differential geometry and topology.
We define a categorical birational invariant for minimal geometrically rational surfaces with a conic bundle structure over a perfect field via components of a natural semiorthogonal decomposition. Together with the similar known result on…
We develop a universal framework to study smooth higher orbifolds on the one hand and higher Deligne-Mumford stacks (as well as their derived and spectral variants) on the other, and use this framework to obtain a completely categorical…
We give a new description of the data needed to specify a morphism from a scheme to a toric Deligne-Mumford stack. The description is given in terms of a collection of line bundles and sections which satisfy certain conditions. As…
Hypertoric varieties are determined by hyperplane arrangements. In this paper, we use stacky hyperplane arrangements to define the notion of hypertoric Deligne-Mumford stacks. Their orbifold Chow rings are computed. As an application, some…