SISSA Algebraic Geometry Seminar – Spring 2024

This is the webpage of the SISSA Algebraic Geometry Seminar. The topic for the Spring Semester will be Hilbert and Quot schemes. An outline of the seminar is here.

Detailed schedule

- 12 February 2024,
__Nicolò Bignami__(SISSA):*Construction of Quot schemes* - 19 February 2024,
__Lorenzo Barbato__(SISSA):*Hilbert schemes of curves in*ℙ^{3},*twisted cubics* - 26 February 2024,
__Riccardo Moschetti__(Università di Torino):*TBA* - 4 March 2024,
__Pedro Santos__(UFPB, Brazil):*Nested Hilbert schemes via quivers* - 11 March 2024,
__Andrea Ricolfi__(SISSA):*Hilbert schemes and moduli of ideals* - 18 March 2024,
__Christian Forero Pulido__(SISSA):*Cohomology of*Hilb(*S*) and representations of the Heisenberg algebras - Tue 26 March 2024,
__Sergej Monavari__(EPFL, Lausanne):*Tetrahedron instantons in Donaldson-Thomas theory*. – Venue: ICTP, Luigi Stasi Seminar Room at 14:30 - 8 April 2024,
__Michele Graffeo__(SISSA):*Reducibility of*Hilb^{78}(𝔸^{3}) - 15 April 2024,
__Ian Selvaggi__(SISSA):*Motivic invariants of Quot schemes of points* - 22 April 2024,
__Riccardo Ontani__(SISSA):*POT on*Hilb^{n}(*X*)*and degree 0 DT invariants*

- 29 April 2024,
__Søren Gammelgaard__(SISSA):*Projective noncommutative partial resolutions and quiver varieties*

**Abstract**: We introduce a class of "projective noncommutative schemes" related to the action of a finite SL_2(C)-subgroup G on P^2, building on work by Baranovsky, Ginzburg, and Kuznetsov. We show that despite being noncommutative, these objects enjoy useful properties - for instance, a version of Serre duality - and that they can be interpreted as partial resolutions of the singular (commutative!) scheme P^2/G. We also show that these noncommutative schemes can be used to give interpretations of several Nakajima quiver varieties, generalising earlier results. This is a report on ongoing work with Ádám Gyenge.

*Venue*: Room 136 (SISSA) at 16:00. -
TRINO talk
13 May 2024
__Paolo Stellari__(Università di Milano):*Deformations of stability conditions with applications to very general Hilbert schemes of points and abelian varieties*

**Abstract**: The construction of stability conditions on the bounded derived category of coherent sheaves on smooth projective varieties is notoriously a difficult problem, especially when the canonical bundle is trivial. In this talk, I will illustrate a new and very effective technique based on deformations.

A key ingredient is a general result about deformations of bounded t-structures (and with some additional and mild assumptions). Two remarkable applications are the construction of stability conditions for very general abelian varieties in any dimension and for irreducible holomorphic symplectic manifolds of Hilb^n-type, again in all possible dimensions. This is joint work with C. Li, E. Macrì, Alex Perry and X. Zhao.

*Venue*: Room 136 (SISSA) at 16:00. -
TRINO talk
14 May 2024
__Felix Thimm__(University of Oslo):*The 3-fold K-theoretic DT/PT vertex correspondence*

**Abstract**: Donaldson-Thomas (DT) and Pandharipande-Thomas (PT) invariants are two curve counting invariants for 3-folds. In the Calabi-Yau case, a correspondence between the numerical DT and PT invariants has been conjectured by Pandharipande and Thomas and proven by Bridgeland and Toda using wall-crossing. For equivariant K-theoretically refined invariants, the DT/PT correspondence reduces to a DT/PT correspondence of equivariant K-theoretic vertices. In this talk I will explain our proof of the equivariant K-theoretic DT/PT vertex correspondence using a K-theoretic version of Joyce's wall-crossing setup. An important technical tool is the construction of a symmetized pullback of a symmetric perfect obstruction theory on the orginial DT and PT moduli stacks to a symmetric almost perfect obstruction theory on auxiliary moduli stacks. This is joint work with Nick Kuhn and Henry Liu.

*Venue*: Room 136 (SISSA) at 10:00. - 20 May 2024,
__Shubham Sinha__(ICTP):*Counting maps from curves to Grassmannians*

**Abstract**: In this talk, I will construct a virtual fundamental class for the Quot scheme over curves and use it to study the 'virtual' intersection theory of the Quot scheme. I will present the Vafa-Intriligator formula for these intersection numbers and show that they provide an enumerative count of maps from a fixed smooth curve C to Grassmannians, when the degree is large or the genus of C is zero. As an application, I will also derive a ring presentation for the quantum cohomology of Grassmannians. -
TRINO talk
28 May 2024,
__Lars Halle__(Università di Bologna):*Computing the base change conductor for Jacobians*

**Abstract**: Let K be a discretely valued field, with ring of integers R. The base change conductor of an abelian K-variety, denoted c(A), is a numerical invariant which measures the failure of A to have semi-abelian reduction over R. It can be difficult in general to compute c(A) explicitly. In this talk I will present an approach for Jacobians, using intersection theory and invariants of quotient singularities on certain normal R-models of the curve in question. Joint work with D. Eriksson and J. Nicaise.

*Venue*: Room 131 (SISSA) at 15:00.