ICTP/SISSA Research Seminar on Quot schemes – Fall 2024

Barbara Fantechi, Lothar Göttsche, Alina Marian, Shubham Sinha and I organise, since October 2024, a research seminar around the geometry and virtual intersection theory of the Quot scheme of a smooth projective surface. We plan to study the irreducibility of the Quot scheme, the existence of a virtual fundamental class and the relationship between Quot scheme invariants and the Donaldson type invariants arising from the moduli space of sheaves on the surface.
Everyone is very welcome to attend and contribute a talk. For info, please email any of the organisers.

Schedule

We meet every Thursday at 14:30, either at SISSA or at ICTP.

• 10 October 2024, Alina Marian (ICTP): Introduction. Venue: ICTP, Fermi Building, Room 101.
• 17 October 2024, Nicolò Bignami (SISSA): Irreducibility of some Quot schemes on a smooth curve. Venue: SISSA, Room 137.
• 24 October 2024, Nicolò Bignami (SISSA): On some irreducibility questions for Quot schemes and moduli spaces of semistable sheaves on surfaces I. Venue: ICTP, Fermi Building, Room 101.
• 31 October 2024, Nicolò Bignami (SISSA): On some irreducibility questions for Quot schemes and moduli spaces of semistable sheaves on surfaces II. Venue: SISSA, Room 137.
• 14 November 2024, Andrea Ricolfi (SISSA): Virtual classes on Quot schemes. Venue: SISSA, Room 137.
• 21 November 2024, Barbara Fantechi (SISSA): Expanded degenerations, Quot schemes and relative obstruction theories. Venue: Fibonacci Lecture Room, ICTP Galileo Guest house.
  Abstract. A classical technique in algebraic geometry is "if something bad happens in a moduli problem in a one parameter family over the node in a nodal curve, t blow up the mode and hope for best". We will discuss such methods in the context of Quot schemes in Li and Wu’s arXiv:1110.0390, similar work by Marian, Oprea and Pandharipande arXiv:0904.2992, and relevant obstruction theories.
• 28 November 2024, TBA (---): TBA. Venue: SISSA, Room 137.
• 5 December 2024, Joachim Jelisiejew (University of Warsaw): Singularities of Quot(𝔸²) and Hilb(𝔸³) via bicanonical modules. Venue: ICTP, Fermi Building, Room 101.
  Abstract. In the talk I will discuss the singularities (and how to resolve them) of Quot_d((O_𝔸²)^r) and Hilb_d(𝔸³) for very small d, any r. I will also paint a conjectural picture of what can be expected and what is known for larger d. A key new player is the bicanonical module. This is very much a work in progress, joint with Ritvik Ramkumar and Alessio Sammartano.
• 12 December 2024, TBA (---): TBA. Venue: SISSA, Room 137.