Calendar

Bertini's theorem in positive characteristic

Cesar Hilario (IMPA)

Abstract: The Bertini-Sard theorem is a classical result in algebraic geometry. It states that in characteristic zero almost all the fibers of a dominant morphism between two smooth algebraic varieties are smooth; in other words, there do not exist fibrations by singular varieties with smooth total space. Unfortunately, the Bertini-Sard theorem fails in positive characteristic, as was first observed by Zariski in the 1940s. Investigating this failure naturally leads to the classification of its exceptions. By a theorem of Tate, a fibration by singular curves of arithmetic genus g in characteristic p > 0 may exist only if p <= 2g + 1. When g = 1 and g = 2, these fibrations have been studied by Queen, Borges Neto, Stohr and Simarra Canate. A birational classification of the case g = 3 was started by Stohr (p = 7, 5), and then continued by Salomao (p = 3). In this talk I will report on some progress in the case g = 3, p = 2. In fact, a great variety of examples exist and very interesting geometric phenomena arise from them.

Motivic classes of algebraic stacks

Federico Scavia (UBC)

Abstract. The Grothendieck ring of algebraic stacks was introduced by T. Ekedahl in 2009, following up on work of other authors. It is a generalization of the Grothendieck ring of varieties. If G is a linear algebraic group, it is an interesting problem to compute the motivic class of its classifying stack BG in this ring. I will give a brief introduction to the Grothendieck ring of stacks, and then explain some of my results in the area.

Generalized canonical models of foliated surfaces

Yen-An Chen (Utah)

Abstract. By work of McQuillan and Brunella, it is known that foliated surfaces of general type with only canonical foliation singularities admit a unique canonical model. It is then natural to investigate the moduli space parametrizing canonical models. One issue is that the condition being a canonical model is neither open nor closed. In this talk, I will introduce the generalized canonical models to fix this issue and study some properties (boundedness/ separatedness/ properness/ local-closedness) of the moduli space of generalized canonical models.

Moduli spaces of low dimensional abelian varieties with torsion.

Shiva Chidambaram (U Chicago)

Abstract. The Siegel modular variety A_2(3) which parametrizes abelian surfaces with split level 3 structure is birational to the Burkhardt quartic threefold. This was shown to be rational over Q by Bruin and Nasserden. What can we say about its twist A_2(\rho) for a Galois representation \rho valued in GSp(4, F_3)? While it is not rational in general, it is unirational over Q by a map of degree at most 6, showing that \rho arises as the 3-torsion of infinitely many abelian surfaces. In joint work with Frank Calegari and David Roberts, we obtain an explicit description of the universal object over a degree 6 cover using invariant theoretic ideas. Similar ideas work for (g,p) = (1,2), (1,3), (1,5), (2,2), (2,3) and (3,2). When (g,p) is not one of these six tuples, we discuss a local obstruction for representations to arise as torsion.

The S_n-equivariant rational homology of the tropical moduli spaces Delta_{2,n}

Claudia Yun (Brown)

Abstract. The tropical moduli space Delta_{g,n} is a topological space that parametrizes isomorphism classes of n-marked stable tropical curves of genus with total volume 1. Its reduced rational homology has a natural structure of S_n-representations induced by permuting markings. In this talk, we focus on Delta_{2,n} and compute the characters of these S_n-representations for n up to 8. We use the fact that Delta_{2,n} is a symmetric Delta-complex, a concept introduced by Chan, Glatius, and Payne. The computation is done in SageMath.

Multiplicative Quantum Cobordism Theory

Irit Huq-Kuruvilla (UC Berkeley)

Abstract: K-theoretic Gromov-Witten invariants were proposed by Kontsevich in the 80s, and the foundations were developed by YP Lee in 1999. I will introduce a modified form of these invariants obtained by twisting the virtual structure sheaf by an arbitrary characteristic class of the tangent bundle of the moduli space of stable maps, and state a formula relating the generating function for these invariants to the unmodified ones. I'll also discuss how these invariants can be used to define Gromov-Witten invariants valued in other complex-oriented cohomology theories, the universal example of which is cobordism theory. This talk is based on work from https://arxiv.org/abs/2101.09305.

Cluster structures on Schubert varieties in the Grassmannian

Melissa Sherman-Bennett (UC Berkeley)

Abstract: Cluster algebras are a class of commutative rings with a (usually infinite) set of distinguished generators, grouped together in overlapping subsets called "clusters." They were defined by Fomin and Zelevinsky in the early 2000s; since their definition, connections have been found to representation theory, Teichmuller theory, discrete dynamical systems, and many other branches of math. I'll discuss joint work with K. Serhiyenko and L. Williams, in which we show that homogeneous coordinate rings of Schubert varieties in the Grassmannian are cluster algebras, with clusters coming from a particularly nice combinatorial source.

Non-torsion Brauer groups

Louis Esser (UCLA)

Abstract: The classical definition of the Brauer group of a field can be extended in different ways to general schemes. I'll explain two methods of doing so in order to motivate the question: when is the cohomological Brauer group torsion? After reviewing some techniques for computing this group, I'll present new examples of normal surfaces in positive characteristic with non-torsion Brauer group. This talk is based on work from https://arxiv.org/abs/2102.01799.

Semi-polarized meromorphic Hitchin and Calabi-Yau integrable systems

Jia-Choon Lee (U Penn)

Abstract: Since the seminal work of Hitchin, the moduli spaces of Higgs bundles, also known as the Hitchin systems, have been studied extensively because of their rich geometry. In particular, each of these moduli spaces admits the structure of an algebraic integrable system. There is another class of algebraic integrable systems provided by the so-called non-compact Calabi-Yau integrable systems. By the work of Diaconescu, Donagi and Pantev, it is shown that Hitchin systems are isomorphic to certain Calabi-Yau integrable systems. In this talk, I will discuss joint work with Sukjoo Lee on how to extend this correspondence to the meromorphic setting.

Intersection theory on moduli of hyperplane arrangements and marked del Pezzo surfaces

Nolan Schock (University of Georgia)

Abstract: This talk is about the intersection theory of two of the first examples of compact moduli spaces of higher-dimensional varieties: the log canonical compactification of the moduli space of marked del Pezzo surfaces, and the stable pair compactification of the moduli space of hyperplane arrangements. The latter space is the natural higher-dimensional version of \bar{M_{0,n}} , the moduli space of -pointed rational curves, but its geometry can in general be arbitrarily complicated. On the other hand, the former space, which can also be viewed as a higher-dimensional generalization of \bar{M_{0,n}}, by construction has nice geometry on the boundary, and this leads (conjecturally for degree 1,2) to a presentation of its Chow ring entirely analogous to Keel's famous presentation of the Chow ring of \bar{M_{0,n}}. I will describe work in progress using the relationships between these moduli spaces in order to describe the intersection theory of the moduli space of stable hyperplane arrangements.

Uhlenbeck compactification as a Bridgeland moduli space

Tuomas Tajakka (University of Washington)

Abstract: In recent years, Bridgeland stability conditions have become a central tool in the study of moduli of sheaves and their birational geometry. However, moduli spaces of Bridgeland semistable objects are known to be projective only in a limited number of cases. After reviewing the classical moduli theory of sheaves on curves and surfaces, I will present a new projectivity result for a Bridgeland moduli space on an arbitrary smooth projective surface, as well as discuss how to interpret the Uhlenbeck compactification of the moduli of slope stable vector bundles as a Bridgeland moduli space. The proof is based on studying a determinantal line bundle constructed by Bayer and Macrì. Time permitting, I will mention some ongoing work on PT-stability on a 3-fold.

Characterizations of multigraded regularity on products of projective spaces

Lauren Cranton Heller (UC Berkeley)

Abstract: Eisenbud and Goto described the Castelnuovo-Mumford regularity of a sheaf on projective space in terms of three different properties of the corresponding graded module: its betti numbers, its local cohomology, and its truncations. For the multigraded generalization of regularity defined by Maclagan and Smith, these three conditions are no longer equivalent. I will discuss some relationships between them for sheaves on products of projective spaces.

On the interaction of normal square-tiled surfaces and group theory

Andrea Thevis (RWTH Aachen University)

Abstract: A translation surface is obtained by taking finitely many polygons in the Euclidean plane and gluing them along their edges by translations. If we restrict to gluing unit squares, we obtain a square-tiled surface, also known as origami. In the first part of the talk, I explain some motivations for studying translation surfaces. I especially aim to point out why it is natural to study square-tiled surfaces in some of these contexts. In the second part of the talk, we consider certain square-tiled surfaces with maximal symmetry group in more detail. More precisely, we examine their types of singularities and their Veech groups using group theoretic methods. This is partially joint work with Johannes Flake.

On K3 surfaces admitting symplectic automorphism of order 3

Yulieth Katterin Prieto Montañez (Università di Bologna)

Abstract: The theory of K3 surfaces with symplectic involutions and their quotients is now a well-understood classical subject thanks to foundational works of Nikulin, Morrison, and van Geemen and Sarti. In this talk, we will try to develop analogous results for K3 surfaces with symplectic automorphisms of order three: we will explicitly describe the induced action of these automorphisms on the K3-lattice, which is isometric to the second cohomology group of a K3 surface; we deduce the relation between the families that admitting these automorphisms and the ones given by their quotients. If time permits, we give some applications: one related to Shioda-Inose structures, and another one in the construction of infinite towers of isogeneous K3 surfaces. This is joint work with Alice Garbagnati.

Bertini's theorem in positive characteristic

Cesar Hilario (IMPA)

Abstract: The Bertini-Sard theorem is a classical result in algebraic geometry. It states that in characteristic zero almost all the fibers of a dominant morphism between two smooth algebraic varieties are smooth; in other words, there do not exist fibrations by singular varieties with smooth total space. Unfortunately, the Bertini-Sard theorem fails in positive characteristic, as was first observed by Zariski in the 1940s. Investigating this failure naturally leads to the classification of its exceptions. By a theorem of Tate, a fibration by singular curves of arithmetic genus g in characteristic p > 0 may exist only if p <= 2g + 1. When g = 1 and g = 2, these fibrations have been studied by Queen, Borges Neto, Stohr and Simarra Canate. A birational classification of the case g = 3 was started by Stohr (p = 7, 5), and then continued by Salomao (p = 3). In this talk I will report on some progress in the case g = 3, p = 2. In fact, a great variety of examples exist and very interesting geometric phenomena arise from them.

Social Event

Event link (SpatialChat) and password will be shared via email on 12/14. Please register by Monday to receive this information!

Quantum K-theory of Incidence Varieties

Weihong Xu (Rutgers University)

Abstract. Certain rational enumerative geometry problems can be formulated as intersection theory in the moduli space of stable maps M̅_{0,m}(X,d). This moduli space is well-behaved when X is a projective homogeneous variety G/P. Non-trivial relations among solutions to these enumerative geometry problems (Gromov-Witten invariants) enable the definition of an associative product and in turn a formal deformation of the cohomology ring called the quantum cohomology ring of X. Similarly, a deformation of the Grothendieck ring K(X) called the quantum K-theory ring of X is defined using K-theoretic versions of Gromov-Witten invariants.

After introducing relevant background, we will focus on the quantum K-theory of the projective homogeneous variety Fl(1,n-1;n) (also called an incidence variety), where I have found explicit multiplication formulae and computed some K-theoretic Gromov-Witten invariants. These computations lead to suspected rationality properties of some natural subvarieties of M̅_{0,m}(X,d).

Birational automorphisms and movable cone of Calabi-Yau complete intersections

José Yáñez (University of Utah)

Abstract. In 2013 Cantat and Oguiso used Coxeter groups to calculate the birational automorphism group and prove the Kawamata-Morrison conjecture for varieties of Wehler type. In this talk, we use generalized geometric representations of Coxeter groups to compute the movable cone and to extend Cantat-Oguiso's result to Calabi-Yau complete intersections in products of projective spaces.

Cubic Threefolds and Vanishing Cycles on its Hyperplane sections

Yilong Zhang (Ohio State University)

Abstract. For a general cubic threefold, a vanishing cycle on a smooth hyperplane section is an integral 2-class perpendicular to the hyperplane class with self-intersection equal to -2. The question is what is a vanishing cycle on a singular hyperplane section? We will show that there is a certain moduli space parameterizing "vanishing cycles" on all hyperplane sections and the boundary divisor answers the question. As a vanishing cycle on a smooth cubic surface is represented by the difference of two skew lines, such moduli space arises as a quotient of the Hilbert scheme of skew lines on the cubic threefold. Based on the Abel-Jacobi map on cubic threefolds studied by Clemens and Griffiths, we'll show that the moduli space is isomorphic to the blowup of the theta divisor of the at an isolated singularity.

Top weight cohomology of A_g

Gwyneth Moreland (Harvard University)

Abstract. I will discuss recent work on computing the top weight cohomology of A_g for g up to 7. We use combinatorial methods coming from the relationship between the top weight cohomology of A_g and the homology of the link of the moduli space of tropical abelian varieties to carry out the computation. This is joint work with Madeline Brandt, Juliette Bruce, Melody Chan, Margarida Melo, and Corey Wolfe.

Orbifold Gromov–Witten theory of complete intersections

Nawaz Sultani (University of Michigan)

Abstract. For genus 0 GW invariants of schemes, one can compute the GW theory of a complete intersection in projective space in terms of the GW theory of the ambient space through the so-called Quantum Lefschetz theorem (QL). However, this theorem doesn't necessarily hold when one considers stacky targets, which makes such examples much more difficult to understand.

In this talk, I will discuss the failure of QL in the orbifold case, and present techniques that allow us to compute the g=0 GW invariants in these cases when the target is a complete intersection in a stacky GIT quotient. The work presented is joint with Felix Janda and Yang Zhou. I will also not assume you know anything about GW theory prior.

Hyperplane sections and Moduli

Lisa Marquand (Stony Brook University)

Abstract. One way to produce new varieties from a fixed subvariety of projective space is to intersect with linear subspaces. When we consider a cubic threefold X in P^4, we can consider hyperplane sections: to every hyperplane (considered as a point in the dual projective space) we can associate a cubic surface namely the intersection X ∩ H. One natural question is to ask, given a cubic surface Y, how many times does it appear as a hyperplane section of X (up to projective equivalence)? More rigorously, we can define a rational map which takes a hyperplane H to the class of the intersection, considered as a point in the moduli space of cubic surfaces (GIT). One can check that this is a generically finite surjective map, and thus answering our question is equivalent to calculating the degree of this map. Although the question is enumerative, the techniques involved are particularly interesting: the wonderful blow-up technique of De Concini-Procesi, plus the dual perspective of the moduli of cubic surfaces. This is a work in progress, and we will actually consider a slight modification resulting in easier computations.

q-bic Hypersurfaces

Raymond Cheng (Columbia University)

Abstract. One of the funny features of geometry in positive characteristic is that equations behave of lower degree than they seem. In this talk, I would like to convince you that Fermat hypersurfaces of degree q + 1, q a power of the ground field characteristic, is geometrically analogous to quadric and cubic hypersurfaces. To me, this example suggests a theme with which to understand some geometric features in positive characteristic, like the unexpected abundance of rational curves in certain varieties.

Orlov's Theorem for Smooth Proper Varieties

Noah Olander (Columbia University)

Abstract. Orlov proved in 1996 that many functors between derived categories of smooth projective varieties are represented by kernels, i.e., complexes on the product. Since then, Orlov's theorem has had a profound influence on algebraic geometry. In this talk, we discuss Orlov's proof as well as some technical advances and new ideas which shed light on it, leading to an extension of the theorem to the smooth proper case.

Topology of tropical moduli spaces of weighted stable curves in higher genus

Shiyue Li (Brown University)

Abstract. Tropical moduli spaces of weighted stable curves are moduli spaces of metric weighted marked graphs satisfying certain stability conditions. The space of tropical weighted curves of genus g and volume 1 is the dual complex of the divisor of singular curves in Hassett's moduli space of weighted stable genus g curves. One can derive plenty of topological properties of the Hassett spaces by studying the topology of these dual complexes. In this talk (and in a paper coming soon), we show that the spaces of tropical weighted curves of genus g and volume 1 are simply-connected for all genus greater than zero and all rational weights, under the framework of symmetric Delta-complexes and via a result by Allcock-Corey-Payne 19. We also calculate the Euler characteristics of these spaces and the top weight Euler characteristics of the classical Hassett spaces in terms of the combinatorics of the weights.

Cox rings, linear blow-ups and the generalized Nagata action

Lei Yang (Northeastern University)

Abstract. Nagata gave the first counterexample to Hilbert's 14th problem on the finite generation of invariant rings by actions of linear algebraic groups. His idea was to relate the ring of invariants to a Cox ring of a projective variety. Counterexamples of Nagata's type include the cases where the group is G_a^m for m=3, 6, 9 or 13. However, for m=2, the ring of invariants under the Nagata action is finitely generated. It is still an open problem whether counterexamples exist for m=2.

In this talk we consider a generalized version of Nagata's action by H. Naito. Mukai envisioned that the ring of invariants in this case can still be related to a cox ring of certain linear blow-ups of P^n. We show that when m=2, the Cox rings of this type of linear blow-ups are still finitely generated, and we can describe their generators. This answers the question by Mukai.

P=W phenomena from Fano/LG correspondence

Sukjoo Lee (University of Pennsylvania)

Abstract. P=W phenomena, originated from non-abelian Hodge theory, has been recently formulated by A.Harder, L.Katzarkov and V.Przyjalkowski in the context of mirror symmetry of log Calabi-Yau manifolds. In particular, if the log Calabi-Yau manifold admits Fano compactification (X,D) with smooth anti-canonical divisor D, we can study P=W phenomena from categorical viewpoint under the Fano/LG correspondence. In this talk, we will go over the story and generalize to the case where D has more than one component.

Bott vanishing using GIT and quantization

Sebastián Torres (UMass Amherst)

Abstract. A smooth projective variety is said to satisfy Bott vanishing if Ω^j ⊗ L has no higher cohomology for every j and every ample line bundle L. This is a very restrictive property, and there are few non-toric examples known to satisfy it. I will present a new class of examples obtained as smooth GIT quotients of P^n. For this, I will need to use the work by Teleman and Halpern-Leistner about the derived category of a GIT quotient, and explain how this allows us, in some cases, to compute cohomologies directly in an ambient quotient stack.

A generic talk on irrationality

Nathan Chen (Stony Brook University)

Abstract. Given a smooth projective variety, there are two natural questions that can be asked: (1) How can we determine when it is rational? and (2) If it is not rational, can we measure how far it is from being rational? There has been a great deal of recent progress towards developing invariants with the second question in mind. We will explain some new techniques involved in bounding these invariants for certain classes of varieties.

Fourier-Mukai theory for stacky genus 1 curves

Libby Taylor (Stanford University)

Abstract. We will discuss a theory of derived equivalences for certain Artin stacks. We will apply this theory to study the derived categories of genus 1 curves and of their Picard stacks. Some questions we will answer: when are two G_m gerbes over genus 1 curves derived equivalent? If C and C' are derived equivalent curves, can we prove that C' is the moduli space of certain vector bundles on C? If C'=Pic^d(C), is it true that C=Pic^f(C') for some f, and if so, can we use Fourier-Mukai theory to find f? (Spoilers: when one is Pic^d of the other; yes; yes and yes.) This is joint work with Soumya Sankar.