AG Seminar (past talks)

Utrecht University

In Block 3 (Spring 2024), the seminar will take place on Thursday at 10-11am in different locations. For the seminar lunch, we meet at 11:50 on the ground floor of the HFG. 

Woonam Lim (Utrecht) -- Moduli spaces of one dimensional sheaves on the projective plane

Abstract: Moduli spaces of one dimensional sheaves on the projective plane have been studied in connections to enumerative geometry and meromorphic Hitchin system. Cohomology group of the moduli space is equipped with a perverse filtration which plays an important role in these connections. I will explain how to study perverse filtration by combining ideas from tautological ring, BPS integrality and chi-independence. I will finish by proposing a conjectural algebraic characterization of the perverse filtration. This is a joint work in progress with Y. Kononov, M. Moreira, W. Pi. 


Dylan Butson (Oxford) -- Perverse coherent extensions on Calabi-Yau threefolds and representations of cohomological Hall algebras

Abstract: Motivated by the problem of generalizing the ADHM construction of the moduli space of framed torsion free sheaves on P^2, to describe the entire stack of quiver representations geometrically and moreover to extend this to some other toric surfaces, I'll explain a construction of certain moduli stacks of coherent sheaves on toric Calabi-Yau threefolds which admit a natural equivalence with stacks of representations of a framed quiver with potential, following results of Bridgeland and Van den Bergh. I'll also describe natural representations on the critical cohomology groups of subvarieties of stable objects in these stacks, towards proving a generalization of a conjecture of Alday-Gaiotto-Tachikawa.


Reinier Schmiermann (Utrecht University) -- Murphy's law on a fixed locus of the Quot scheme

Abstract: The Quot and Hilbert schemes of 0 dimensional sheaves on A^d are moduli spaces that are known to be highly singular for sufficiently large values of d. In particular, starting from d = 16, by a result of Jelisiejew they are known to satisfy a form of "Murphy's law", meaning that they have arbitrarily bad singularities. It is however still unknown what is the smallest value of d for which we can expect such pathological behaviour. In an attempt to answer this question, in this talk we will study the singularities of the locus of the Quot scheme consisting of fixed points under the torus action coming from A^d. In particular, I will show that already for d = 4 this locus satisfies Murphy's law.


Younghan Bae (Utrecht University) -- Generalized Beauville decomposition

Abstract: By the work of Beauville and Deninger-Murre, the rational cohomology group (in fact, the relative Chow motive) of abelian scheme over a regular base has a canonical decomposition into pure weight pieces. The essential ingredient is the Fourier-Mukai transformation relative to the base. It is a natural question to ask what happens if there exist singular fibers. For a family of integral local planar curves, the relative compactified Jacobian is an example of such “degenerate abelian fibration”. In this talk, we consider perverse filtration on the rational cohomology of the relative compactified Jacobian and ask when the filtration has Fourier-stable multiplicative splitting. If the relative compactified Jacobian arise from Beauville-Mukai system, we get such multiplicative splitting. On the other hand, for general family of integral curves, even for family of nodal curves, such multiplicative splitting cannot exist. This is a joint work with D. Maulik, J. Shen and Q. Yin.


Ana María Botero (University of Bielefeld) -- Toroidal b-divisors and applications in differential and arithmetic geometry

Abstract: We define toroidal b-divisors on a quasi projective variety. These can be seen as conical functions on a balanced polyhedral space. We show the existence of an intersection pairing for so called nef toroidal b-divisors, which gives rise to a Monge-Ampére type measure on the polyhedral space. We then show some applications of this theory. On the one hand side, b-divisors are used to encode singularities of psh metrics and we derive Chern-Weil type formulae for such metrics on line bundles. On the other hand, using a Hilbert-Samuel formula, we compute asymptotic dimension formulae of spaces of automorphic forms on mixed Shimura varieties. If time permits, we connect our work to the notion of adelic line bundles of Yuan and Zhang, and outline some current research directions.



Ratko Darda (Université de Paris (IMJ-PRG)) -- The Manin conjecture for toric stacks

Abstract: A basic question in the Diophantine geometry is how many integer solutions to a system of polynomial equations are there, or, in other words, how many rational points are there on algebraic varieties. When there are “many” solutions, the precise asymptotic behaviours are predicted by the Manin conjecture. Recently, the Manin conjecture has been generalized to Deligne–Mumford stacks. Recall that the Deligne–Mumford stacks are the algebro-geometric objects which arise as solutions to the questions of parametrizing objects having finite (not necessarily trivial) automorphism groups. In this talk, we study the Manin conjecture for toric stacks, which are Deligne–Mumford stacks, whose geometry can be described using combinatorial data similar to that describing toric varieties.


Younghan Bae (Utrecht University) -- Intersection theory on compactified Jacobians over the moduli spaces of stable curves

Abstract: I will give three talks on compactified Jacobians over the moduli space of stable curves. Three talks will be independent and loosely related. 

Over the moduli space of stable curves, integrals of tautological classes have been an important subject. For example, Witten-Kontsevich theorem says that the generating series of  integrals of \psi classes satiesfies the KdV hierarchy.  What can we say about relative compactified Jacobian? Using the quasi-stable model, there is a natural choice of tautological classes on compactified Jacobian. In this talk, we will see that the pushforward along the forgetful morphism from the compactified Jacobian to the moduli space of stable curves preserves tautological classes. Our main ingredient is the universal double ramification cycle formula. Using the Witten-Kontsevich theorem, one can compute integrals of tautological classes on compactified Jacobian. Along the way, I will explain how this idea can be adapted to study the logarithmic Picard group constructed by Molcho-Wise. This is a joint work in progress with S. Molcho.


Younghan Bae (Utrecht University) -- Fourier transform and class of sections

Abstract: Fourier transformation is an important tool to study the Chow group (or relative Chow motive) of abelian schemes. By Arinkin, the Fourier transforamtion extends to relative compactified Jacobian for family of integral local planar curves. Using Arinkin’s Fourier transformation, we will see how to compute the class of Abel-Jacobi sections on the relative Jaocbian over a moduli space of integral nodal curves. This computation recovers Pixton’s formula without r-polynomial and partially recovers the universal double cycle formula by Bae-Holmes-Pandharipande-Schmitt-Schwartz. This is a joint work with S. Molcho.


Sara Mehidi (Utrecht University) -- The Logarithmic Jacobian and extension of torsors over families of degenerating curves

Abstract: Molcho and Wise constructed the log Picard group, a canonical compactification of the Picard group of families of nodal curves that is smooth, proper and  possesses a group structure, but which can only be represented in the category of logarithmic spaces. Afterwards, it was shown that the restriction of this log Picard group to the degree zero log line bundles - which gives the so called Logarithmic Jacobian-  is in fact the log Néron model of the Jacobian of the smooth locus. In this talk, I will first explain the construction of this material. Then, I will show that the Logarithmic Jacobian classifies finite log torsors over families of nodal curves, generalizing the classical situation for smooth curves. Finally, we will see that the Néron property of the Log Jacobian allows to get a result on extending fppf torsors into log torsors over families of nodal curves. This is a joint work with Thibault Poiret.


Younghan Bae (Utrecht University) -- Generalized Faber-Zagier relations on relative Jacobian

Abstract: Let M_g be the moduli space of smooth genus g algebraic curves. By Madsen-Weiss, the rational cohomology of M_g is a free algebra generated by tautological classes in the “stable range”. Outside the “stable range”, there are relations among tautological classes. Faber-Zagier conjectured that there is a structure of relations among tautological classes outside the “stable range”. This conjecture, and its extension to the moduli space of stable curves, is proven by Pandharipande-Pixton-Zvonkine and Janda. Similar stability results extends to the relative Jacobian over M_g by Ebert-Randal-Williams. It is an interesting question to ask whether we have a generalised Faber-Zagier relations on the relative Jacobian. I will give a candidate of those relations using the stable quotients over the relative Jacobian. This is a joint work in progress with H. Lho.


Andrea Ricolfi (SISSA) -- The motive of the Hilbert scheme of points 

Abstract: The geometry of Hilbert schemes of points is largely unknown, or known to be pathological in a precise sense. This should in principle make most (naive) invariants essentially inaccessible. In this talk we explain how to obtain a closed formula for the generating function of the motives (classes in the Grothendieck ring of varieties) of Hilbert schemes of points Hilb(X,d) for X a smooth variety of arbitrary dimension, and for fixed number of points d < 9. Work in progress with M. Graffeo, S. Monavari and R. Moschetti.



In Block 2 (Fall 2023), the seminar will take place on Wednesdays at 13:45-14:45 in BBG 001 or BBG 017, after lunch, unless otherwise stated.

Felix Thimm (University of Oslo) -- Nekrasov's formula for some orbifolds

Abstract: Nekrasov's formula computes equivariant K-theoretic DT invariants for Hilbert schemes of points of toric CY3-folds. We extend this to certain global quotient CY3 orbifolds, refining a result of Young to equivariant K-theoretic DT theory and proving a conjecture by Cirafici. Okounkov's proof of Nekrasov's formula combines two techniques. First, the virtual structure sheaves satisfy a certain factorization property, which allows us to simplify the general form of the DT generating series. Secondly, the rigidity principle allows us to determine the remaining unknowns by computing a limit in the equivariant parameters. We will explain these techniques in the case of schemes and describe some of the modifications to make them work for orbifolds.


Yannik Schuler (Sheffield) -- Equivariant Strings and Gromov-Witten theory

Abstract: In the past three decades some of the most exciting developments in enumerative geometry were inspired by analogies in mathematical physics. After giving you a crash course on how an algebraic geometer should think about (A-model topological) string theory, I will make a proposal for a mathematically rigorous formulation of the so called refined topological string in the framework of equivariant Gromov-Witten theory on Calabi-Yau fivefolds. Moreover, I will present a surprising correspondence between the so called Nekrasov-Shatashvili limit for a local surface K_S and the enumeration of tangents to a smooth anti-canonical curve in S (assuming the latter exist). This is ongoing work with Andrea Brini.


Tudor Padurariu (Max Planck Bonn) -- Quasi-BPS categories for K3 surfaces

Abstract: Hyperkahler varieties are higher dimensional analogues of K3 surfaces. Moduli spaces of semistable sheaves on a K3 surface for a primitive Mukai vector are examples of hyperkahler varieties. Other hyperkahler varieties are obtained as crepant resolutions of such moduli spaces when the Mukai vector is not primitive. However, it is known that there is only one new example produced in this way, constructed by O’Grady. 

In joint work with Yukinobu Toda, we construct dg categories which are analogues of crepant resolutions of singularities for the moduli space of semistable sheaves on a K3 surface for a generic stability condition and a general Mukai vector. The construction is inspired by the study of BPS invariants in enumerative geometry of Calabi-Yau threefolds.







Noah Arbesfeld (Vienna) -- Computing vertical Vafa-Witten invariants

Abstract: I'll present a computation in the algebraic approach to Vafa-Witten invariants of projective surfaces, as introduced by Tanaka-Thomas. The invariants are defined using moduli spaces of stable Higgs pairs on surfaces and are formed from contributions of components. The physical notion of S-duality translates to conjectural symmetries between these contributions. One component, the "vertical" component, is a nested Hilbert scheme on a surface. I'll explain work in preparation with M. Kool and T. Laarakker in which we express invariants of this component in terms of a quiver variety, the instanton moduli space of torsion-free framed sheaves on P^2. As a consequence, we deduce constraints on Vafa-Witten invariants, including a formula for the contribution of the vertical component to refined invariants in rank 2.


Leo Herr (Leiden) -- Concerning Weil restrictions

Abstract: We begin with a zoo of examples of Weil restrictions. The main three are: 1) number theoretic, 2) jet spaces, and 3) the Kontsevich space of (pre)stable maps. We discuss each in their own right, leading to an elementary main theorem which applies to all of them. Open questions and doable exercises are emphasized throughout, and input encouraged.


Ariyan Javanpeykar (Nijmegen) -- Non-density of rational points on algebraic varieties over number fields

Abstract: Which varieties over a number field should have a potentially dense set of rational points?  A naive guess based on Lang's conjectures was made in the early nineties: a smooth projective variety over a number field which does not dominate a positive-dimensional variety of general type after any etale covering should have a dense set of rational points over some large enough number field. This conjecture is probably false (and the "correct" conjecture was later formulated by Campana). In joint work with Finn Bartsch and Erwan Rousseau we disprove  the natural analogue  of the aforementioned naive conjecture for transcendental-rational points (a notion I will explain in this talk).  In our proof, we develop the theory of moduli spaces of orbifold maps, establish an analogue of Kobayashi-Ochiai's finiteness theorem for dominant maps in Campana's orbifold setting, employ Mori's bend-and-break and ultimately rely on Faltings's finiteness theorem (formerly Mordell's conjecture).



2023

2022

2021