Tag - Resolution of singularities

Alberto San Miguel Malaney: Partial Resolutions of Affine Symplectic Singularities II

9th September, 2024

We will continue to discuss partial resolutions of conical affine symplectic singularities, particularly their deformation theory and Springer theory. First we will explain the construction of the universal deformations of symplectic singularities and their partial resolutions, generalizing the Grothendieck-Springer resolution. Then we will use these universal deformations to study the Springer theory of symplectic singularities and their partial resolutions, using recent work of McGerty and Nevins. In particular, we will compute the cohomology of the fibres of the partial resolutions under suitable conditions, generalizing a result of Borho and MacPherson for the nilpotent cone. Finally, we will use partial resolutions to construct and study symplectic resolutions of symplectic leaf closures, generalizing the Springer maps from cotangent bundles of partial flag varieties to nilpotent orbit closures.

Alberto San Miguel Malaney: Partial Resolutions of Affine Symplectic Singularities I

26th August, 2024

Symplectic singularities are a generalization of symplectic manifolds that have a symplectic form on the smooth locus but allow for certain well-behaved singularities. They have a strong relationship to representation theory and include nilpotent cones of semisimple Lie algebras, quiver varieties, affine Grassmannian slices, and Kleinian singularities. There is a combinatorial description for partial resolutions of conical affine symplectic singularities, stemming from Namikawa's 2013 result that a symplectic resolution is also a relative Mori Dream Space. In this talk we will explore these partial resolutions in more detail, exploring their birational geometry, deformation theory, and Springer theory. In particular, we will review the definition of the Namikawa Weyl group for conical affine symplectic singularities and use birational geometry to define a generalization for their partial resolutions. We will also use this Namikawa Weyl group to classify the Poisson deformations of the partial resolutions. We will then describe how these partial resolutions fit into the framework of Springer Theory for symplectic singularities, following Kevin McGerty and Tom Nevins' recent paper, Springer Theory for Symplectic Galois Groups. Finally, we will discuss some ongoing research that stems from these ideas, inspired by parabolic induction and restriction.

Kenji Iohara: On Elliptic Root Systems

20th February, 2023

In 1985, K. Saito introduced elliptic root systems as root systems belonging to a real vector space F equipped with a symmetric bilinear form I with signature (l, 2, 0). Such root systems are studied in view of simply elliptic singularities which are surface singularities with a regular elliptic curve in its resolution. K. Saito had classified elliptic root systems R with its one-dimensional subspace G of the radical of I, in the case when R/G ⊂ F/G is a reduced affine root system. In our joint work with A. Fialowski and Y. Saito, we have completed its classification; we classified the pair (R,G) whose quotient R/G ⊂ F/G is a non-reduced affine root system. In this talk, we give an overview of elliptic root systems and describe some of the new root systems we have found.

Francesca Carocci: The geometry of moduli spaces of stable maps to projective space and modular desingularisation via log blowups

7th September, 2020

An LMS online lecture course in moduli spaces.

Moduli spaces of stable maps have been of central interest in algebraic geometry for the last 30 years. In spite of that, the geometry of these spaces in genus bigger than zero is poorly understood, as the Kontsevich compactifications include many components of different dimensions meeting each other in complicated ways, and the closure of the smooth locus is difficult to describe.

In recent years a new perspective on the problem of finding better behaved compactifications, ideally smooth ones, has come from log geometry. This approach has proved successful in a series of examples and log geometry is now becoming a natural setting to study modular resolutions of moduli spaces.

The aim of this series of talks will be to see how log geometry techniques apply to give modular smooth compactifications of moduli spaces of stable maps to projective spaces in genus one and two; we will also explain why the latter are interesting from an enumerative point of view.

In more detail: we begin by studying the deformation theory and the global geometry of moduli spaces of genus one and two stable maps; we then give a brief introduction to log schemes, line bundles on log schemes and log blowups and conclude by exhibiting the log modification resolving the moduli spaces of maps in genus one and two and explaining their modular meaning.