Schedule
Monday [05.06.2023]
09:15 - 10:30   Larson
11:00 - 12:15   Canning
Lunch at 12:30
16:30 - 18:15   Mellit I
Tuesday [06.06.2023]
09:15 - 10:30   Pixton
11:00 - 12:15   Schmitt
Lunch at 12:30
16:30 - 18:15   Mellit II
Wednesday [07.06.2023]
09:15 - 10:30   Marian
11:00 - 12:15   Farkas I
Lunch at 12:30
Afternoon free
Thursday [08.06.2023]
09:15 - 10:30   Kennedy-Hunt
11:00 - 12:15   Farkas II
Lunch at 12:30
16:30 - 18:15   Bérczi
Friday [9.06.2023]
09:30 - 10:45   Blanc
Titles and Abstracts
G. Bérczi
Tautological intersection theory of Hilbert scheme of points
While the Hilbert scheme of points on surfaces is pretty well-understood, the Hilbert scheme over manifolds presents a mixture of pathological and unknown behaviour: our knowledge of their components, singularities and deformation theory is very limited.
After a brief survey we report on a new approach to calculate tautological intersection numbers of the main component and of certain geometric subsets which play crucial role in enumerative geometry applications. We present a Chern-Segre-type positivity conjecture for tautological integrals coming from global singularity theory.
J. Blanc
Birational maps of Severi-Brauer surfaces, with applications to Cremona groups of higher rank
We describe the group of birational transformations of a non-trivial Severi-Brauer surface over a perfect field K, proving in particular that if it contains a point of degree 6, then it is not generated by elements of finite order as it admits a surjective group homomorphism to ℤ.
We then use this result to study Mori fibre spaces over the field of complex numbers, for which the generic fibre is a non-trivial Severi-Brauer surface. We prove that any group of cardinality at most the one of ℂ is a quotient of any Cremona group of rank at least 4. As a consequence, this gives a negative answer to the question of Dolgachev of whether the Cremona groups of all ranks are generated by involutions. We also prove that the 3-torsion of the Cremona group of rank at least 4 is not countable. Joint work with J. Schneider and E. Yasinsky.
S. Canning
Semi-tautological systems and the cohomology of the moduli space of curves
I will introduce the notion of semi-tautological systems, which are systems of subalgebras with a minimal set of functoriality properties of the cohomology rings of the moduli spaces of stable curves. They are designed to study the structure of the cohomology of the moduli spaces of stable curves beyond the tautological ring. I will give a criterion for a given semi-tautological system to span all of cohomology in a given degree. Using this criterion and other results about the moduli space of curves, both topological and algebro-geometric, I will explain new results about cohomology of high degree. This is joint work with Hannah Larson and Sam Payne.
G. Farkas I
The birational geometry of the moduli of curves
via tropical geometry and non-abelian Brill-Noether theory
I will discuss how novel ideas from non-abelian Brill-Noether theory can be used to prove that the moduli space of genus 16 is uniruled. For the much studied question of determining the Kodaira dimension of M_g, this case has long been understood to be crucial in order to make further progress. This is joint work with Verra.
G. Farkas II
The Minimal Resolution Conjecture on points on generic curves
The Minimal Resolution Conjecture predicts the shape of the resolution of general sets of points on a projective variety in terms of the geometry of the variety. We present an essentially complete solution to this problem for general curves. Our methods also provide a proof (valid in arbitrary characteristic) of Butler's Conjecture on the stability of syzygy bundles on a general curve of every genus at least 3, as well as of the Frobenius semistability in positive characteristic of the syzygy bundle of a general curve in the range d>2r-1. Joint work with E. Larson.
P. Kennedy-Hunt
The logarithmic Hilbert scheme and its tropicalisation
A basic question is understanding how the Hilbert/ Quot scheme of a projective variety X changes when we degenerate X. The key to answering this question is restricting attention to subschemes/ sheaves that are transverse to a simple normal crossing divisor. I will explain how to construct compact moduli spaces called the logarithmic Hilbert/Quot schemes which track this transverse geometry. A key part of the story is a related tropical moduli problem reminiscent of a tropical version of a Hilbert scheme. This discussion is a sheaf-theoretic parallel to the theory of logarithmic stable maps and generalises the spaces appearing in logarithmic Donaldson-Thomas theory. It will lead to new enumerative geometric invariants, such as Quot scheme invariants on logarithmic surfaces. A future hope is to understand how other moduli spaces of coherent sheaves and their stability conditions behave under degeneration.
H. Larson
Chow rings of moduli spaces of stable curves
The moduli space Mbar_{g,n} of n-pointed stable genus g
curves possesses certain natural classes known as tautological
classes. The main question I'll address in this talk is: for which g,
n is the Chow ring of Mbar_{g,n} generated by tautological classes? We
prove that this holds for several new values of g and n. In these
cases, the cycle class map is an isomorphism and the moduli spaces
have polynomial point count. This is joint work with Samir Canning.
A. Marian
Tautological integrals on the moduli space of stable bundles on a curve and a Segre-Verlinde correspondence
I will discuss the older result that any tautological integral on the moduli space of stable bundles of rank r on a smooth projective curve C can be expressed as a sum of integrals on r-products of Sym^d(C) for suitable degrees, therefore reducing the calculation to rank one. This uses the Quot scheme on C. I will give a recent application of the method: a Verlinde-Segre correspondence for the moduli space of stable bundles of coprime rank and degree. It is interesting to keep a parallel perspective on the geometry of stable sheaves on surfaces.
A. Mellit
Tautological algebras of surfaces I and II
In the first talk, I will describe an algebra which acts on the cohomology of any "sufficiently nice" moduli space associated to a smooth (compact or not) algebraic surface. For those who know, the algebra is in fact the 0-dimensional COHA extended by the tautological ring, but we won't need this description. The algebra is very close to the trigonometric Cherednik algebra and has the following features: 1) description in terms of generators and quadratic and cubic relations. 2) Fock representation. I will explain these.
In the second talk, I will explain how in the case when the surface is the total space of a line bundle on a curve the algebra degenerates and acquires an extra "perverse grading", which is basically the contents of the P=W conjecture. If time permits, I will also explain the proof of the Segre-Verlinde correspondence, which does not use the algebra per se, but hopefully will eventually be put in the same context.
A. Pixton
Primitive tautological relations
The tautological ring of the moduli space of stable curves is a subring of the Chow ring consisting of the cycles produced by forgetful and gluing morphisms. Relations in this ring can be pulled back to give recursion relations in Gromov-Witten theory. I will define the notion of a "primitive" tautological relation, one that cannot be derived from simpler relations using certain basic operations. I will discuss how to construct many primitive relations in each genus g (quadratically many in g).
J. Schmitt
Using computer algebra to study intersection theory of moduli spaces
One way to gain information on the cohomology groups of moduli
spaces of curves is via computer experiments. In the talk I will start
with a brief introduction to the SageMath package "admcycles", which
implements the tautological ring of these moduli spaces as well as many
known geometrically defined cycle classes. In the main part of the talk,
I give a tour of past results and ongoing projects that used this
package to calculate enumerative invariants, prove negative results and
explore conjectural relations between cycle classes. Time permitting, I
also sketch some work in progress and future directions for improving
the package. Audience members are encouraged to bring their own laptops to try out
some of the calculations themselves!