Warwick Algebraic Geometry Seminar

Abstracts

Hamid Ahmadinezhad (Loughborough University) - Birational Rigidity of Fano 3-folds in Higher Codimensions

I will give an overview of birational rigidity for Fano 3-folds. I will explain what it means and what we know. I will then discuss this notion in codimensions 2 and 3 and its connection to (un)projections.

Timothy Logvinenko (Cardiff University) - P-Functors and Cyclic Covers

I will begin by reviewing the geometry of a cyclic cover branched in a divisor. I will then explain how it gives us the first ever example of a non-split P-functor. This is a joint work with Rina Anno (Kansas).

Hülya Argüz (Imperial College London) - Real Lagrangians in Toric Degenerations

One of the main tools of the Gross-Siebert program in mirror symmetry is toric degenerations constructed from integral affine manifolds with singularities. The real loci of such degenerations provide interesting examples of Lagrangians which conjecturally are amenable to algebraic-geometric versions of Floer theory. In this talk I will discuss how the topology of the real locus can be understood by means of affine geometry and by Kato-Nakayama spaces associated to log spaces. This talk reports on joint work with Bernd Siebert.

David Ssevviiri (Makerere University, Kampala, Uganda) - Completely Prime Modules and 2-Primal Modules

A notion of prime ideal in a commutative ring can be expressed in many equivalent statements that become distinct when the ring is assumed to be noncommutative. This leads to: completely prime ideals, strictly prime ideals, strongly prime ideals, s-prime ideals, l-prime ideals, etc. Each of the aforementioned "prime" has a module analogue; and these analogues collapse to just one notion when a module is defined over a commutative ring. In this talk, I discuss the completely prime (sub)modules, their properties and torsion theories induced by the completely prime radical. Secondly, by comparing completely prime (sub)modules with prime (sub)modules, I talk about a class of 2-primal modules which is a module analogue of 2-primal rings.

Ben Davison (University of Glasgow) - The Donaldson-Thomas-Lie Algebra

The theory of noncommutative Donaldson-Thomas invariants is a local model for the theory of DT invariants on Calabi-Yau 3-folds. I will explain recent progress in categorifying this local model, and in particular the presence of a surprising decomposition theorem and perverse filtration for the cohomology of the moduli stacks appearing in the theory. Via this perverse filtration we obtain a Lie algebra structure on the space of cohomological BPS invariants, generalising the construction of Borcherds-Kac-Moody Lie algebras associated to quivers.

Andreas Gross (Imperial College London) - Chi-y Genera of Generic Intersections in Algebraic Tori and Refined Tropicalizations

An algorithm to compute chi-y genera of generic complete intersections in algebraic tori has already been known since the work of Danilov and Khovanskii in 1978, yet a closed formula has been given only very recently by Di Rocco, Haase, and Nill. In my talk, I will show how this formula simplifies considerably after an extension of scalars. I will give an algebraic explanation for this phenomenon using the Grothendieck rings of vector bundles on toric varieties . We will then see how the tropical Chern character gives rise to a refined tropicalization, which retains the good properties of the usual, unrefined tropicalization.

Gwyn Bellamy (University of Glasgow) - Symplectic Resolutions of Quiver Varieties

Quiver varieties, as introduced by Nakaijma, play a key role in representation theory. They give a very large class of symplectic singularities and, in many cases, their symplectic resolutions too. However, there seems to be no general criterion in the literature for when a quiver variety admits a symplectic resolution. In this talk I will give necessary and sufficient conditions for a quiver variety to admit a symplectic resolution. This result builds upon work of Crawley-Boevey and of Kaledin, Lehn and Sorger. The talk is based on joint work with T. Schedler.

Álvaro Nolla de Celis (Universidad Rey Juan Carlos, Madrid) - On G-Hilb for Dihedral Groups in SL(3,C)

In this talk I will show how can we calculate G-Hilb(C³) explicitly when G is a dihedral (intransitive) group in SL(3,C). I will explain with this case the main features and difficulties appearing when considering non-abelian group actions and how combining G-sets and representations of the McKay quiver will lead to an answer. The work is in progress.

Daniel Evans (University of Liverpool) - Birationally Rigid Complete Intersections of Codimension Two

Birational (super) rigidity is known for almost all families of Fano complete intersections of index one in the projective space. Typically birational superrigidity was shown for a generic (in particular non-singular) variety in the family. Now improved techniques make it possible to allow certain types of singularities. In this talk I will discuss the case of codimension two complete intersections. This is a joint work with Aleksandr Pukhlikov.

Miles Reid (University of Warwick) - Trihedral Groups, Boats and G-Hilb C³

I will explain how to compute an affine covering of GHilb C³. Each affine piece consists of clusters with a given set of monomials forming a "nearly monomial" basis of the quotient ring O_Z. As Alvaro explained last week, sorting out the nearly monomial bases is a clean computational way of determining the lacings of the McKay quiver for G, whereas the noncommutative algebra associated to quivers is a technique that writes out a priori the ideal of equations defining each affine piece. This long-term program of work has several collaborators, and you may have heard previous iterations of the talk.

Isabel Stenger (Technische Universität Kaiserslautern) - Computing Numerical Godeaux Surfaces with no Torsion

Numerical Godeaux surfaces provide the first case in the geography of minimal surfaces of general type. It is known that the torsion group of such a surface is cyclic of order m ≤ 5 and a full classification has been given for m = 3, 4, 5 by Reid, Miyaoka. In my talk I will discuss a homological approach to construct a numerical Godeaux surface X based on a former project of Frank-Olaf Schreyer. The main idea is to study the syzygies of the canonical ring R(X) considered as a module over some (weighted) graded polynomial ring. We focus on the case Tors(X) = 0 and pay particular emphasis to the genus-4 fibration given by the bicanonical system.

Angela Gibney (University of Georgia) - On Finite Generation of the Section Ring of the Determinant Line Bundle

I will discuss recent work, with Prakash Belkale, where we show the section ring for the pair (Bun, D) is finitely generated, for D the determinant of cohomology line bundle on the stack Bun = Bun_SL(r)(C) parameterizing principal SL(r)-bundles on a singular stable curve C. I'll define these things, put the result into some historical perspective, and mention two or three applications, depending on the time.

Nikolaos Tziolas (University of Cyprus) - The Automorphism Scheme of Smooth Canonically Polarized Surfaces in Positive Characteristic

Let X be a smooth canonically polarized surface defined over an algebraically closed field of characteristic p>0. In this talk I will present some results about the geometry of X in the case when the automorphism scheme Aut(X) of X is not smooth, or equivalently X has nontrivial global vector fields. This is a situation that appears only in positive characteristic and is intimately related to the structure of the moduli stack of canonically polarized surfaces in positive characteristic. One of the results that will be presented in this talk is that smooth canonically polarized surfaces with non-smooth automorphism scheme and “small” invariants are algebraically simply connected and uniruled.