This was the webpage of the Algebraic Geometry Seminar at Utrecht University in 2023-2024. See here for the current informations about AG seminar at UU.
In Block 4 (Spring 2024), the seminar will usually take place on Wednesday at 1-2pm. For the seminar lunch, we meet at 11:50 on the ground floor of the HFG.
April 24 (13:00-14:00, BBG 083) [This week on April 26 also: Belgian Dutch Algebraic Geometry Seminar]
Ruijie Yang (Berlin) -- Minimal exponent of a hypersurface
Abstract: Recently, the minimal exponent of a hypersurface over complex numbers has been understood as a quite useful refined invariant of the log canonical threshold. It has found many new applications including deformation of Calabi-Yau 3-folds (Friedman-Laza), higher rational and higher du Bois singularities (Mustata-Popa) and geometric Schottky problem (Schnell-Yang). However, some properties of this invariant remain mysterious. In this talk I will discuss the conjecture of Mustata and Popa on birational characterization of the minimal exponent via log resolutions, which is the main obstruction for the computation in practice. I will explain the heuristic of the Mustata-Popa conjecture from Igusa's work on counting integer solutions of congruence equations and the monodromy conjecture. Then I will discuss how several ideas from Hodge theory and geometry representation theory can lead to a better understanding of the minimal exponent (birationally). This is based on the joint work with Christian Schnell on higher multiplier ideals and the joint work in progress with Dougal Davis on a new description of the Kashiwara-Malgrange V-filtration of mixed Hodge modules.
May 1 (13:00-14:00, BBG 161)
Felix Wierstra (Utrecht) -- The Mumford conjecture
Abstract: The Mumford conjecture is an important result that computes the cohomology of the moduli space of curves in the stable range and was originally proved by Madsen and Weiss about 20 years ago. Recently, a new alternative proof of the Mumford conjecture was given by Bianchi. In this talk, I will discuss a version of Bianchi's proof which was further streamlined by Das and Petersen.
May 8 (13:00-14:00, BBG 083)
May 15 (13:00-14:00, BBG 083)
Bianca Gouthier (University of Bordeaux) -- Infinitesimal rational actions
Abstract: For any finite $k$-group scheme $G$ acting rationally on a $k$-variety $X$, if the action is generically free then the dimension of $Lie (G)$ is upper bounded by the dimension of the variety. This inequality turns out to be also a sufficient condition for the existence of such actions, when $k$ is a perfect field of positive characteristic and $G$ is infinitesimal commutative trigonalizable. These group schemes are non-reduced and arise only in positive characteristic. After presenting the main objects involved and overviewing the motivation for this problem, we willl explain the result in the case of actions of the $p$-torsion of a supersingular elliptic curve.
May 21 (10:00-11:00, KBG - ATLAS) -- unusual day, time, location
Rahul Pandharipande (ETH Zurich) -- Higher genus GW theory of the Hilbert scheme of points of the plane
Abstract: The quantum cohomology of the Hilb(C2,n) is determined by the operator of quantum multiplication by the divisor class (and was computed 20 years ago in work with Okounkov). I will explain how to think about the higher genus theory from several perspectives: CohFT, MNOP, and, in the case of genus 1, the intersection theory of the moduli space A_g of PPAVs. The talk is connected to recent joint and disjoint work of several mathematicians: S. Canning, F. Greer, A. Iribar Lopez, C. Lian, S. Molcho, D. Oprea, A. Pixton,and H.-H. Tseng.
May 29 (13:00-14:00, BBG 083)
Young-hoon Kiem (KIAS) -- Shifted Lagrange multipliers method
Abstract: Lagrange multipliers method relates critical points on a submanifold with those on the ambient manifold. In derived algebraic geometry, we are allowed to consider a more general type of functions called shifted functions and their critical loci. In this talk, I will discuss how Lagrange multipliers method adapted to derived algebraic geometry naturally produces quantum Lefschetz principle of Chang-Li and more.
June 05 (13:00-14:00, BBG 083)
Thibault Poiret (University of St Andrews) -- Moduli of roots of universal line bundles
Abstract: Let L be a line bundle on the universal curve over M_{g,n}. The moduli space S^0 of r-th roots of L admits a natural compactification S over the Deligne-mumford compactification of M_{g,n}. When r is 2 and L is the canonical bundle, S^0 is the well-studied moduli space of spin curves, which is known to have two connected components: one parametrizing even spin curves, and one parametrizing odd spin curves. I will describe a natural partition of S into connected locally closed subspaces for arbitrary r and L. This partition is compatible with the partition of the Deligne-Mumford compactification indexed by stable graphs. This is joint work in progress with Margarida Melo.
June 12 (13:00-14:00, BBG 083)
Raju Krishnamoorthy (Humboldt Universität Berlin) -- Rank 2 local systems and abelian varieties
Abstract: Motivated by work of Corlette-Simpson over the complex numbers, we conjecture that all irreducible rank 2 ell-adic local systems with cyclotomic determinant and infinite image on a smooth variety over a finitely generated field come from families of abelian varieties. We will survey partial results on this conjecture. Time permitting, we will provide indications of the proofs, which involve the work of Hironaka and Hartshorne on positivity, Drinfeld's first work on the Langlands correspondence over function fields, space filling curves, and the pigeonhole principle. The results are all joint work with either Ambrus Pál or Jinbang Yang and Kang Zuo.
June 19 (13:00-14:00, BBG 201)
Katharina Hübner (Goethe-Universität Frankfurt) -- An integral structure on the sheaf of differentials
Abstract: We explain how to define an integral structure on the sheaf of differentials $\Omega_X$ of an adic space $X$. This should be thought of as an analogue of the subsheaf $\mathcal{O}_X^+$ of the structure sheaf $\mathcal{O}_X$. This integral structure $\Omega_X^+$ can be described in terms of logarithmic differentials on a log regular model (if such a model exists). Possibly assuming resolution of singularities this gives a strategy for transferring results on log étale cohomology to tame cohomology.
June 26 (13:00-14:00, Minnaert building Room 0.14) -- unusual location
Lucien Hennecart (University of Edinburgh) -- Cohomological integrality isomorphisms
Abstract: The theory of cohomological Hall algebras has proven powerful in studying Donaldson-Thomas invariants of some 3-Calabi-Yau categories. It is in particular crucial to obtain cohomological integrality identities. Roughly speaking, the cohomological integrality results concern the finiteness of some invariants associated to the category. They also give cohomologically refined invariants. In my talk, I will give a brief overview of such results, explain their meaning, and explain how to obtain them in various contexts. I will concentrate on a new situation given by symmetric representations of reductive groups.
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.
February 8 (10:00-11:00, BBG 115)
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.
February 15 (10:00-11:00, HFG 409)
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.
February 22 (10:00-11:00, BBG 061)
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.
February 29 (10:00-11:00, BBG 061)
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.
March 7 (10:00-11:00, BBG 061)
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.
March 14 (10:00-11:00, BBG 061) [This week also: Enumerative Geometry and Arithmetic]
March 21 (10:00-11:00, HFG 409)
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.
March 26 (10:00-11:00, Minnaert building Room 0.13) -- unusual day and location
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.
March 28 (10:00-11:00, BBG 061)
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.
April 4 (10:00-11:00, KBG 224)
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.
April 11 (10:00-11:00, BBG 017) [This week also: DIAMANT Symposium]
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.
April 18 (10:30-11:30, BBG 017) -- unusual time [This week also: (infinity,n)-categories and their applications]
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.
November 15 (13:45-14:45, BBG 001) [This week also: Intercity Number Theory Seminar in Leiden]
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.
November 22 (13:45-14:45, BBG 001) [This week also: DIAMANT Symposium]
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.
November 29 (13:45-14:45, BBG 001)
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.
December 06 - no talk - [MI heidag]
December 13 (13:45-14:45, BBG 017) [This week: Intercity Number Theory Seminar in Amsterdam]
December 20 (13:45-14:45, BBG 001)
December 27 - no talk
January 03 - no talk
January 10 (13:45-14:45, BBG 001)
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.
January 17 (13:45-14:45, BBG 001)
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.
January 24 (13:45-14:45, BBG 001)
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).
January 31 (13:45-14:45, BBG 017)
2023
November 07 (11:15-12:15, BBG 017) Gerard van der Geer (UvA-Beijing-Luxembourg) The cycle class of the supersingular locus
Abstract: Deuring gave a formula for the number of supersingular elliptic curves in characteristic p. We generalize this to a formula for the cycle class of the supersingular locus in the moduli space of principally polarized abelian varieties of given dimension g. The formula determines the class up to a multiple and shows that it lies in the tautological ring. We also give the multiple for g up to 4. This is joint work with S. Harashita.
October 31 (11:15-12:15, BBG 161) Caleb Springer (University College London) Doubly isogenous genus-2 curves over finite fields
Abstract: Our main question is the following: Can you tell the difference between two curves defined over a finite field if you only know their zeta functions and the zeta functions of certain covers? This distinguishing problem is partially motivated by a possible strategy for developing a deterministic polynomial-time algorithm for factoring polynomials over finite fields due to Kayal and Poonen. Ultimately, this talk is concerned with cases where the distinguishing problem is not solvable due to the existence of "very similar" curves. Specifically, we study a family of genus-2 curves with a dihedral action and show that so-called doubly isogenous pairs of curves are surprisingly common in this family. We also provide an explanation of this phenomena which corrects the naive heuristics. This is joint work with Arul, Booher, Groen, Howe, Li, Matei and Pries.
October 24 (11:15-12:15, HFG 409) Denis Nesterov (Vienna) Unramified Gromov-Witten and Gopakumar-Vafa invariants
Abstract: Kim, Kresch and Oh defined moduli spaces of unramified stable maps, which are natural generalisations of (compactified) Hurwitz spaces for a target of an arbitrary dimension. Just like Hurwitz spaces, which are smooth irreducible varieties after normalisation, moduli spaces of unramified stable maps are 'better' compactifications than moduli spaces of stable maps. Pandharipande conjectured that unramified Gromov-Witten invariants of a projective threefold are equal to Gopakumar-Vafa (BPS) invariants in the case of Fano classes (classes that intersect negatively with the canonical class) and primitive Calabi-Yau classes (trivial intersection). After a gentle introduction to unramified Gromov-Witten theory, we will discuss a work in progress which aims to prove the conjecture. This provides a geometric construction of Gopakumar-Vafa invariants in these cases. The proof is based on a certain wall-crossing technique.
October 10 (11:15-12:15, HFG 409) Shivang Jindal (Edinburgh) 2d Cohomological Hall Algebras for Cyclic Quivers and and Integral form of Affine Yangian.
Abstract: In 2012, Schiffmann and Vasserot considered a Hall algebra type construction on the cohomology of moduli of sheaves supported on points on a plane and used it to prove AGT conjecture. However due to the mysterious nature of the moduli of representations of pre-projective algebra, these algebras are very hard to study and are often highly non trivial. They are conjectured to be the same as Maulik-Okounkov Yangians which has further applications in Quantum Cohomology. In this talk, my goal is to give an introduction to these algebras and explain how one can use tools from Cohomological DT theory to study these algebras. In particular, I will explain how for the case of cyclic quiver, this algebra turn out to be a half of the universal enveloping algebra of the Lie algebra of matrix differential operators on torus, while its deformation turn out be an explicit integral form of Affine Yangian of gl(n).
September 28 (9:15-10:15, BBG - 223) Annette Huber-Klawitter (Freiburg) Periods and o-minimality
Abstract: Periods are numbers obtained by integrating algebraic differential forms over semi-algebraic domains. The set contains interesting numbers like logarithms of algebraic numbers or the values of the Riemann zeta function at integral points. The linear or algebraic relations between them is a classical topic of transcendence theory. Grothendieck gave a more conceptual interpretation in terms of the pairing between singular and algebraic de Rham cohomology of algebraic varieties over the rationals. This leads to the (wide open) Period Conjecture predicting all relations between periods. Kontsevich and Zagier suggested to extend the theory to the so called exponential periods appearing in the theory of irregular connections. The conceptual part of the story has been worked out by Hien and Fresan-Jossen. In joint work with Johan Commelin an Philipp Habegger we show that all exponential periods can be written as volumes in a certain o-minimal structure. This hints at a deeper connection between periods and o-minimiality.
September 19 (11:15-12:15, HFG 610) Dusan Dragutinovic (Utrecht) Ekedahl-Oort types of stable curves
Abstract: In this talk, I will present some invariants of curves in positive characteristic p, such as the p-rank, the a-number, or the Ekedahl-Oort type, and discuss intrinsic ways to define them. The main focus will be on Moonen's definition of Ekedahl-Oort types of smooth curves in terms of Hasse-Witt triples. I will show that we can extend this definition to all stable curves. The description we obtain in this manner enables us to compute the dimensions of certain loci of curves. Finally, I will mention some new examples in characteristics p = 2 and p = 3.
[This week also: Intercity Number Theory Seminar in Utrecht]
September 12 (11:15-12:15, HFG 610) Carolina Tamborini (Utrecht) Hodge theory and projective structures on compact Riemann surfaces
Abstract: A projective structure on a compact Riemann surface is an equivalence class of projective atlases, i.e. an equivalence class of coverings by holomorphic coordinate charts such that the transition functions are all Moebius transformations. Any compact Riemann surface admits two canonical projective structures: one coming from uniformization's theorem, and one from Hodge theory. These yield two (different) families of projective structures over the moduli space Mg of compact Riemann surfaces. We wish to compare them and give a characterization of the Hodge theoretic family.
September 5 (11:15-12:15, BBG 109) Nicoló Piazzalunga (Rutgers) 4G Networks
Abstract: I'll introduce the equivariant K-theoretic Donaldson-Thomas theory for toric Calabi-Yau fourfolds, and construct its four-valent vertex with generic plane partition asymptotics. String-theoretically, this is the count of BPS states of a system of D0-D2-D4-D6-D8-branes in the presence of a large Neveu-Schwarz B-field. The talk is based on 2306.12995, 2306.12405 and ongoing work.
June 28 (11:30-12:30, HFG 409) Sarah Arpin (Leiden) Adding level structure to supersingular elliptic curve isogeny graphs
Abstract: The classical Deuring correspondence provides a roadmap between supersingular elliptic curves and the maximal orders which are isomorphic to their endomorphism rings. Building on this idea, we add the information of a cyclic subgroup of prime order N to supersingular elliptic curves, and prove a generalisation of the Deuring correspondence for these objects. We also study the resulting ell-isogeny graphs supersingular elliptic curve with level-N structure, and the corresponding graphs in the realm of quaternion algebras. The structure of the supersingular elliptic curve ell-isogeny graph underlies the security of a new cryptographic signature protocol, SQISign, which is proposed to be resistant against both classical and quantum attack.
June 23 (14:00-15:00, HFG 409) Aline Zanardini (Leiden University) Pencils of plane cubics revisited
Abstract: In recent joint work with M. Hattori we have considered the problem of classifying linear systems of hypersurfaces (of a fixed degree) in some projective space up to projective equivalence via geometric invariant theory (GIT). And we have obtained a complete and explicit stability criterion. In this talk I will explain how this criterion can be used to recover Miranda's description of the GIT stability of pencils of plane cubics.
June 21 (11:30-12:30, BBG 165) Lou van den Dries (University of Illinois at Urbana-Champaign) Transseries and Hardy Fields
Abstract: Transseries are formal series involving exp and log. The differential field of transseries is a universal domain for algebraic differential equations with asymptotic side conditions (in analogy with the field of complex numbers being a universal domain for algebraic geometry). I will then apply this to Hardy fields. Until recently there were only very limited ways of extending Hardy fields, but recently we have achieved a more or less complete overview. I will start from scratch and define/explain transseries and Hardy fields, including examples. (Joint work with Matthias Aschenbrenner and Joris van der Hoeven.)
June 14 (11:30-12:30, HFG 409) Gergely Berczi (Aarhus) Tautological intersection theory of Hilbert scheme of points
Abstract: 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 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.
June 7 : (11:30-12:30, HFG 409) Jeongseok Oh (Imperial College, KIAS) Koszul factorisations of moduli spaces and invariants
Abstract: Invariants on moduli spaces are defined to be integrations over the virtual fundamental classes or Euler characteristics of the virtual structure sheaves. Roughly a virtual structure sheaf is a Koszul resolution of the structure sheaf and the virtual fundamental class is its Chern character (multiplied by Todd class). In fact their first algebraic constructions by Li-Tian and Behrend-Fantechi look different. Here we explain these two are the same in a reasonable circumstance. Furthermore, we explain the use of Koszul factorisations rather than Koszul resolutions gives rise to Kiem-Li localisations of Li-Tian and Behrend-Factechi constructions. These are joint works with Bumsig Kim and Bhamidi Sreedhar. We can use these to produce invariants of moduli spaces of stable sheaves on Calabi-Yau 4-folds as explained in the joint work with Richard Thomas.
May 24 (11:30-12:30, HFG 409) Nancy Abdallah (University of Borås) Lefschetz properties of Artin Gorenstein algebras in low codimensions
Abstract: The weak and strong Lefschetz properties (WLP and SLP) has been much studied for Artinian algebras. Codimension two Artinian algebras over a field of characteristic zero have the strong Lefschetz property. It is open whether Artin Gorenstein (AG) algebras of codimension c=3 satisfy the weak or strong Lefschetz properties. Harima, Migliore, Nagel, and Watanabe proved that complete intersection algebras of codimension three satisfy the weak Lefschetz property. For c=4, Gondim showed that WLP always holds for regularity r <= 4 and gives a family where WLP fails for any r >= 7, building on an earlier example of Ikeda of failure for r=5. In this talk we go through an overview of Lefschetz properties over AG algebras and we give some recent results in codimensions 3 and 4. In codimension 4 we explore relations between Lefschetz properties of an AG algebra A and the free resolutions of A. Lefschetz properties of the Milnor algebras of projective hypersurfaces will be discussed.
May 17 (11:30-12:30, HFG 409) Samir Canning (ETH) Semi-tautological systems and the cohomology of the moduli space of curves
Abstract: I will introduce the notion of semi-tautological systems, which are systems of subalgebras with 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 give several applications. These applications include a complete description of the thirteenth cohomology of the moduli space of stable n pointed curves of genus g for all g,n and that all cohomology classes of sufficiently high degree are tautological. This is joint work in progress with Hannah Larson and Sam Payne.
May 10, (11:30-12:30 MIN 2.01) Thorsten Schimannek (CNRS LPTHE) Counting curves on non-Kaehler Calabi-Yau 3-folds with Topological Strings
Abstract: In general, a projective Calabi-Yau threefold with nodal singularities does not admit a Kaehler small resolution. This happens in particular if the exceptional curves are torsion in homology. In this talk we will discuss how the classical relationship between topological string theory, enumerative geometry and mirror symmetry generalizes to this setting. After recalling some of the ideas from the smooth case, we will both explain the physical intuition behind the generalization and translate it into a concrete mathematical proposal. At the end of the talk, if time permits, we will highlight some open questions that follow from this proposal, related to Donaldson-Thomas theory, non-commutative geometry, and FJRW-theory.
May 3, (13.30-14.30, HFG 409) Chia-Fu Yu (Academia Sinica, Taiwan) Arithmetic invariants on supersingular Ekedahl-Oort (EO) strata for g=4
Abstract: The supersingular locus is one of main interests in algebraic geometry in characteristic p, and can be described in terms of polarised flag type quotients (PFTQs) in the sense of Li and Oort. The description for g=3 is rather explicit and is exploited by Karemaker and Yobuko and myself for investigating the arithmetic invariants of supersingular abelian threefolds, namely, the endomorphism rings and automorphism groups of them, confirming Oort's conjecture for g=3. In this talk we shall explain a general method for investigating the arithmetic invariants on supersingular EO strata, and report the progress of the joint work with Karemaker for g=4.
April 26, (13:30-15:30, HFG 409) Minicourse by Carolina Tamborini (UU) Families of Galois covers of the line: examples and construction
April 25, (13:30-15:30, HFG 610) Minicourse by Carolina Tamborini (UU) Some results on totally geodesic subvarieties in the Torelli locus
April 20, (13:30-15:30, HFG 610) Minicourse by Carolina Tamborini (UU) Torelli map, its differential, its second fundamental form, and the Lie bracket map
April 19, (13:30-14:30, HFG-409) research talk by Oliver Leigh (Uppsala) The Blowup Formula for the Instanton Part of Vafa-Witten Invariants on Projective Surfaces
Abstract: In this talk I will present a blow-up formula for the generating series of virtual χ_y-genera for moduli spaces of sheaves on projective surfaces. The formula is related to a conjectured formula for topological χ_y-genera of Göttsche, and is a refinement of a formula of Vafa-Witten relating to S-duality. I will also discuss the proof of the formula, which is based on the blow-up algorithm of Nakajima-Yoshioka for framed sheaves on ℙ^2. This talk is based on joint work with Nikolas Kuhn and Yuuji Tanaka.
April 18, (13:30-15:30, HFG 610) Minicourse by Carolina Tamborini (UU) Siegel space as symmetric space
April 5, (13:30, HFG-409) research talk by Fabian Reede (Hannover) Line bundles on noncommutative algebraic surfaces
Abstract: Let X be a complex algebraic surface. Let A be a (noncommutative) coherent O_X-algebra whose generic stalk is a central division algebra. Then the pair (X,A) can be thought of as a noncommutative algebraic surface. This talk deals with the moduli space of line bundles on (X,A), i.e., of locally free A-modules of rank one, in the cases where X has Kodaira dimension zero or is Fano. It turns out that the moduli space can be compactified by adding torsion-free A-modules of rank one. We will study e.g. the smoothness of this compactification and the deformation theory of the sheaves involved. This is partly joint work with U. Stuhler and with N.Hoffmann.
March 22, (13:30, HFG-409) research talk by Sara Mehidi (Bordeaux) Extending torsors via log schemes
Abstract: We give here an approach of the problem of extending torsors defined on the generic fiber of a family of curves. The question is to extend each of the structural group and the total space of the torsor above the family. This problem has been studied by many researchers, starting by the first ideas of Grothendieck who solved the case of a constant group of order prime to the residual characteristic. When we are interested in algebraic varieties from an arithmetic point of view, it is natural to consider torsors under a finite flat group that is not necessarily constant: we talk about fppf torsors. In fact, we know from the literature that there are cases where the problem does not have a solution in this setting.
So the idea is to look for a solution in a larger category, namely the category of logarithmic torsors. We will show in particular that the existence of such an extension amounts to extending group functors and morphisms between them. Then, we will compute the obstruction for the extended log torsor to lift into an fppf one. Finally, we give an example of a computation of an extension of a torsor over a given hyperelliptic curve as an application of our results.
March 15, (13:30, HFG-409) research talk by Henry Liu (Oxford): Multiplicative vertex algebras and wall-crossing in equivariant K-theory
Abstract: I will give an overview of recent progress in wall-crossing for equivariant K-theoretic invariants of moduli of sheaves, based on a new framework of Joyce. A multiplicative version of vertex algebras plays a central role in this and related stories. I will give some applications to refined Vafa-Witten theory, the 3-fold DT/PT correspondence, and, if time permits, also some speculation about modularity and S-duality.
March 1, (13:30, HFG-409) research talk by Stefano Marseglia (UU): Cohen-Macaulay type of endomorphism rings of abelian varieties over finite fields
Abstract: In this talk, we will speak about the (Cohen-Macaulay) type of the endomorphism ring of abelian varieties over a finite field with commutative endomorphism algebra. We will exhibit a condition on the type of End(A) implying that A cannot be isomorphic to its dual. In particular, such an A cannot be principally polarised or a Jacobian. This is partly joint work with Caleb Springer.
February 15, (13:30, BBG-069) research talk by Andreas Braun (Durham): Hodge classes on Calabi-Yau fourfolds
Abstract: A crucial question in string theory concerns its set of solutions, particularly those giving rise to an effective four-dimensional description. The most general such solutions are found by specifying a Calabi-Yau fourfold equipped with an elliptic fibration, together with a Hodge class. Without assuming any background I will briefly explain how these objects arise in physics, and which constraints they need to obey. After formulating some of the central questions and conjectures that arise in this context, I will discuss some recent progress.
February 8, (13:30, HFG-409) research talk by Hyeonjun Park (KIAS): Cosection localization via derived algebraic geometry
Abstract: Cosection localization is one of the most powerful tools in virtual enumerative geometry. In this talk, we revisit cosection localization from the perspective of derived algebraic geometry. I will explain derived reduction by (-1)-shifted closed 1-forms and localization through homotopical intersection theory. I will also provide an intrinsic description of cosection-localized virtual cycles using (-2)-shifted symplectic structures. This is based on joint works with Younghan Bae and Martijn Kool, with Dhyan Aranha, Adeel Khan, Alexei Latyntsev, and Charanya Ravi, and with Young-Hoon Kiem.
February 1, (11:00, HFG-409) research talk by Olivier de Gaay Fortman (Hannover): Real moduli spaces, unitary Shimura varieties and non-arithmetic lattices
Abstract: Hodge theory can sometimes be used to identify a moduli space of complex varieties with a complex ball quotient, or an open subset of such a space. I will explain that similar things happen for moduli of real varieties. Real moduli spaces of smooth varieties are often not connected, however - to get a connected moduli space one is led to allow some singularities. It turns out that, very similar to the way in which the connected components of the space of smooth varieties embed into the larger moduli space, real arithmetic ball quotients can be glued together to form a large real ball quotient. Unitary Shimura varieties provide the right framework for this glueing procedure. I will explain how this works, constructing non-arithmetic lattices in PO(n,1) for every n.
January 18, (10:00, HFG-409) research talk by Mick van Vliet (UU): Tame geometry and Hodge theory
Abstract: Tame geometry, made precise by the concept of o-minimal structures, has recently led to some interesting developments in algebraic geometry. In the first half of this talk I will motivate and explain the definition of o-minimal structures, and review some remarkable theorems that hold in the resulting framework of tame geometry. In the second half of the talk, based on work of Bakker, Klingler, and Tsimerman (1810.04801), I will give an overview of a recent application of tame geometry to Hodge theory.
January 11, (11:00, HFG-409) research talk by Lars Halvard Halle (University of Bologna): Degenerations of Hilbert schemes and relative VGIT
Abstract: This talk will be a report on joint work with K. Hulek and Z. Zhang. First I will explain how some central results in VGIT can be extended to a relative setting. After this, I will discuss an application of relative VGIT to the study of certain degenerations of Hilbert schemes of points.
2022
December 7, (HFG-409) research talk by Leo Herr (Leiden University): The Rhizomic Topology
Abstract: What is a sheaf on a log scheme X? If we take the ordinary etale topology, we ignore the log structure. Taking the log étale topology, even the structure "sheaf" O_X is not a sheaf! The same goes for M_X, \overline M_X. We introduce a new "rhizomic" topology on log schemes coarser than the log etale topology. Will this be enough?
November 30, (HFG-409) research talk by Navid Nabijou (QMU London): Roots and logs in the enumerative forest
Abstract: Logarithmic and orbifold structures provide two different paths to the enumeration of curves with fixed tangencies to a normal crossings divisor. Simple examples demonstrate that the resulting systems of invariants differ, but a more structural explanation of this defect has remained elusive. I will discuss joint work with Luca Battistella and Dhruv Ranganathan, in which we identify birational invariance as the key property distinguishing the two theories. The logarithmic theory is stable under strata blowups of the target, while the orbifold theory is not. By identifying a suitable system of “slope-sensitive” blowups, we define a “limit" orbifold theory and prove that it coincides with the logarithmic theory. Our proof hinges on a technique – rank reduction – for reducing questions about normal crossings divisors to questions about smooth divisors, where the situation is much-better understood.
November 23, (HFG-409) research talk by Mar Curco Iranzo (UU): Generalised Jacobians of modular curves and their Q-rational torsion
Abstract: The Jacobian J0(N) of the modular curve X0(N) has received much attention within arithmetic geometry for its relation with cusp forms and elliptic curves. In particular, the group of Q-rational points on X0(N) controls the cyclic N-isogenies of elliptic curves. A conjecture of Ogg predicted that, for N prime, the torsion of this group comes all from the cusps. The statement was proved by Mazur and later generalised to arbitrary level N into what we call generalised Ogg’s conjecture. Consider now the generalised Jacobian J0(N)m with respect to a modulus m. This algebraic group also seems to be related to the arithmetic of X0(N) through the theory of modular forms. In the talk we will present new results that compute the Q-rational torsion of J0(N) for N an odd integer with respect to a cuspidal modulus m. These generalise previous results of Yamazaki, Yang and Wei. In doing so, we will also discuss how our results relate to generalised Ogg’s conjecture.
November 16, (HFG-409) research talk by Francesca Carocci (EPFL): BPS invariant from non Archimedean integrals
Abstract: We consider moduli spaces of one-dimensional semistable sheaves on del Pezzo and K3 surfaces supported on ample curve classes. Working over a non-archimedean local field F, we define a natural measure on the F-points of such moduli spaces. We prove that the integral of a certain naturally defined gerbe on the moduli spaces with respect to this measure is independent of the Euler characteristic. Analogous statements hold for (meromorphic or not) Higgs bundles. Recent results of Maulik-Shen and Kinjo-Coseki imply that these integrals compute the BPS invariants for the del Pezzo case and for Higgs bundles. This is a joint work with Giulio Orecchia and Dimitri Wyss.
October 26, (HFG-409) research talk by Reinier Schmiermann: On Classifying Continuous Constraint Satisfaction Problems
Abstract: The computational complexity class of the existential theory of the reals contains problems which can be reduced to checking whether a system of polynomial equations has a solution over the real numbers. The complexity of a lot of problems in computational geometry turns out to be captured by this class (they are complete for this class). These completeness proofs often use the completeness of a specific Continuous Constraint Satisfaction Problem (CCSP) as an intermediate step. We attempt to give a more systematic analysis of the computational complexity of these CCSPs, and show that a large class of CCSPs is complete for the existential theory of the reals. In this talk, I will give a introduction to computational complexity, the existential theory of the reals, and CCSPs. Then I will state our results, and give a sketch of the proof. This talk is based on joint work with Tillmann Miltzow.
October 19, (HFG-409) Carel Faber, preprint talk on "On the Chow and cohomology rings of moduli spaces of stable curves" by Canning and Larson, arXiv:2208.02357
October 12 (HFG-409), research talk by Remy van Dobben de Bruyn: A variety that cannot be dominated by one that lifts
Abstract: The recent proofs of the Tate conjecture for K3 surfaces over finite fields start by lifting the surface to characteristic 0. Serre showed in the sixties that not every variety can be lifted, but the question whether every motive lifts to characteristic 0 is open. We give a negative answer to a geometric version of this question, by constructing a smooth projective variety that cannot be dominated by a smooth projective variety that lifts to characteristic 0.
October 5, research talk by Dusan Dragutinovic: Computing binary curves of genus five
Abstract. In this talk, we will present algorithms used to determine, up to isomorphism over $\F_2$, all genus five curves defined over $\F_2$ (together with the sizes of their $\F_2$-automorphism groups). Furthermore, we will discuss the outcome considering the Newton polygons of computed curves and mention the obtained stack count $|\mathcal{M}_5(\F_2)|$.
September 21, Boaz Moerman, preprint talk on "Weak approximation and the Hilbert property for Campana points" by Nakahara and Streeter, arXiv:2010.12555
July 6 (11:00 - 12:00, HFG-611), research talk by Valentijn Karemaker
June 28 (11:00 - 13:00, HFG-611), research talk by Pol van Hoften
June 22 (11:15, HFG-610) Carolina Tamborini, "Punctual characterization of the unitary flat bundle of weight 1 PVHS and application to families of curves" by González-Alonso and Torelli, arXiv:2101.03153
June 8 (HFG-610), Wilberd van der Kallen, "Frobenius Splittings", arXiv:1208.3100
May 31 (HFG-610), Dirk van Bree, "When are two HKR isomorphisms equal?" by Huang, arXiv:2205.04439
May 25 (HGF-610), Marta Pieropan, "Heights on stacks and a generalized Batyrev-Manin-Malle conjecture" by Ellenberg, Satriano and Zureick-Brown, arXiv:2106.11340
May 9 (Duistermaat), Marta Pieropan, "Global Frobenius liftability I & II" by Achinger, Witaszek and Zdanowicz, arXiv:1708.03777 and arXiv:2102.02788
March 7, Sebastián Carrillo Santana, "Values of zeta-one functions at positive even integers" by Kobayashi and Sasaki, arXiv:2202.11835
Feb. 28, Carolina Tamborini, "The Coleman-Oort conjecture: reduction to three key cases" by Moonen, arXiv:2201.11971
Feb. 14, Boaz Moerman, "Tamagawa measures on universal torsors and points of bounded height on Fano varieties" by Salberger, article, and "Compter des points d'une variété torique" by de la Bretèche, article
Feb. 7, Sergej Monavari, "On the motive of the Quot scheme of finite quotients of a locally free sheaf" by Ricolfi, arXiv:1907.08123
Jan. 24, Dirk van Bree, "Using the internal language of toposes in algebraic geometry" by Blechschmidt, arXiv:2111.03685
2021
Nov. 29 (HFG-611), Reinier Schmiermann, "Components and singularities of Quot schemes and varieties of commuting matrices" by Jelisiejew and Šivic, arXiv:2106.13137
Nov. 22 (KBG-Atlas), Dusan Dragutinovic, "The existence of supersingular curves of genus 4 in arbitrary characteristic" by Kudo, Harashita and Senda, arXiv:1903.08095
Nov. 8, Dirk van Bree, "Unramified division algebras do not always contain Azumaya maximal orders" by Antieau and Williams, arXiv:1209.2216
Nov. 1, Stefano Marseglia, "On matrices of endomorphisms of abelian varieties" by Zarhin, arXiv:2002.00290, and "Lattices in Tate modules" by Poonen and Rybakov, arXiv:2107.06363
Oct. 11, Carel Faber, "A non-hyperelliptic curve with torsion Ceresa class" by Beauville, arXiv:2105.07160, and "A non-hyperelliptic curve with torsion Ceresa cycle modulo algebraic equivalence" by Beauville and Schoen, arXiv:2106.08390
Oct. 4, Marta Pieropan, "Sums of four squareful numbers" by Shute, arXiv:2104.06966