Birational geometry is the subject of classification of algebraic varieties via birational techniques. In this talk we discuss some of the advances in recent years especially around topics such as boundedness, moduli, and generalised spaces.
Tag - Birational geometry
In this talk we discuss a notion of birational equivalence suitable for Poisson affine varieties: namely, that their function fields are isomorphic as Poisson fields. Some very interesting questions on non-commutative birational geometry, such as the Gelfand-Kirillov Conjecture, make perfect sense in the quasi-classical limit, and naturally leads one to consider the Poisson birational class of the algebras they quantize. In this setting, we study the behaviour of Poisson birational equivalence on the quasi-classical limit of rings of differential operators. With this idea we solve a Poisson analogue of Noether's Problem, introduced by Julie Baudry and François Dumas, in a constructive fashion, for essentially all finite symplectic reflection groups. As applications of our method, we show the Poisson rationality of the Generalized Calogero-Moser spaces, introduced by Etingof and Ginzburg in 2002, and surprisngly for this author, all Coloumb branches of 3d, N=4 SUSY gauge theories - an important object in mathematical physics recently given a rigorous formulation by Nakajima in 2015, and later Nakajima, Braverman, Finkelberg in 2016.
I will talk about the recent progress on the DK conjecture connecting birational geometry and the
derived categories, and related conjectures such as DL conjecture, etc. I will also discuss two kinds of
factorizations of birational maps; those into flips, flops and divisorial contractions according to the
minimal model programme, and more traditional factorizations into blow-ups and blow-downs with
smooth centres.
Fano varieties constitute a fascinating class of algebraic varieties that are important in birational
geometry and beyond. On the other hand, studying mild singularities is an indispensable feature of
modern birational geometry. In this talk I will try to explain how one interwines the two subjects to
prove various local and global boundedness statements regarding linear systems on varieties and
families of Fano varieties.
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