The Geometry of Hilbert Schemes of Points
May 6-10, 2024 BellaVista Relax Hotel, Levico Terme (TN), Italy
The Hilbert scheme is a classically studied moduli space, which parametrises closed subschemes with prescribed Hilbert polynomial in an ambient projective scheme. These moduli spaces where initially introduced by Grothendieck, but turned out to have a prominent rôle in many modern areas outside the realm of Algebraic Geometry: Mathematical Physics, Combinatorics, Theoretical Physics (just to mention some recent examples). Generalisations of the Hilbert scheme have a wide range of applications as well, such as nested Hilbert schemes (parametrising flags of closed subschemes), moduli spaces of framed sheaves and Quot schemes (parametrising quotient sheaves).
The easiest example one can consider is the Hilbert scheme of points, which parametrises closed zero-dimensional subschemes of fixed length. Already in this case, the geometry of the moduli space is rich and intriguing, as it is in general considerably pathological. For instance, if the dimension of the ambient variety is 3, the Hilbert scheme of points is in general reducible, and for dimension larger than 4 its singularities can be as bad as possible.
Hilbert schemes (and their generalisation) played an important role in the development of modern Enumerative Geometry. In fact, for a large and interesting class of cases, the Hilbert scheme of points is not smooth but carries a virtual fundamental class. Donaldson-Thomas theory deals precisely with the (virtual) invariants one can compute on the Hilbert schemes with respect to its virtual structure. Donaldson-Thomas theory is very rich and admits several layers of refinements, for example: cohomological, K-theoretical, elliptic, categorical, motivic and is predicted to match Gromov-Witten invariants.
The conference aims to bring together experts on the topic with the purpose of creating fruitful collaborations. For this reason, during the days of the conference, there will be moments dedicated to mathematical discussion and others dedicated to the presentation of open problems.
SPEAKERS
Alessandra Bernardi Università degli Studi di Trento, IT
Gergely Bérczi Aarhus Universitet, DK
Nadir Fasola University Of Sheffield, UK
Anthony Iarrobino Northeastern University, US
Joachim Jelisiejew Uniwersytet Warszawski, PL
Tomasz Mańdziuk Texas A&M University, US
Cristina Manolache University Of Sheffield, UK
Alina Marian Northeastern University, US - ICTP, IT
Rosa Maria Miró-Roig Universitat de Barcelona, ES
Andrei Negut MIT, US
Ritvik Ramkumar Cornell University, US
Ilaria Rossinelli École Polytecnique Federale de Lausanne, CH
Reinier F. Schmiermann Univeristeit Utrecht, NL
Roy Skjelnes KTH, SE
The conference is organized by
Michele Graffeo SISSA
Paolo Lella Politecnico di Milano
Sergej Monavari EPFL
Andrea T. Ricolfi SISSA
Alessio Sammartano Politecnico di Milano
The conference is supported by
PRIN 2020 "Squarefree Gröbner degenerations, special varieties and related topics" Grant number 2020355B8Y
CIRM Centro Internazionale per la Ricerca Matematica
National Group for Algebraic and Geometric Structures, and their Applications GNSAGA INdAM
PRIN 2022 "Geometry of algebraic structures: moduli, invariants, deformations" Grant number 2022BTA242
Dipartimento di Matematica Politecnico di Milano