Date | Speaker | Title | Abstract |
Tuesday, April 6, 2021 Start: 11:00 AM Location: Online via Zoom (Please email the organizers if interested.) |
Javier Gonzalez-Anaya UC Riverside | A review of the theory of varieties. | This is an introductory talk for the graduate students who will attend the seminar this quarter. We will review the theory of varieties: their construction, the Zariski topology and maps between them. We'll provide examples along the way to illustrate these concepts. Slides |
Tuesday, April 13, 2021 Start: 11:00 AM Location: Online via Zoom (Please email the organizers if interested.) |
Rohini Ramadas Brown University | Dynamics on the moduli space M_{0,n}. | A rational function f(z) in one variable determines a self-map of P^1. A rational function is called post-critically finite (PCF) if every critical (ramification) point is either pre-periodic or periodic. PCF rational functions have been studied for their special dynamics, and their special distribution within the moduli space of all rational maps. By works of W. Thurston and S. Koch, every PCF map (with a well-understood class of exceptions) arises as an isolated fixed point of an algebraic dynamical system on the moduli space M_{0,n} of point-configurations on P^1; these dynamical systems are called Hurwitz correspondences. I will introduce Hurwitz correspondences and their connection to PCF rational maps, and discuss how the dynamical complexity of Hurwitz correspondence can be studied via combinatorial compactifications of M_{0,n}. |
Tuesday, April 20, 2021 Start: 11:00 AM Location: Online via Zoom (Please email the organizers if interested.) |
Julie Rana Lawrence University | T-singular surfaces of general type. | We explore the moduli space of stable surfaces, where the simplest of questions continue to remain open for almost all invariants. A few such questions: Of the allowable singularities, which ones actually occur on a stable surface? Which of these deform to smooth surfaces? How can we use this knowledge to find divisors in the moduli spaces? Can we develop a stratification of these moduli spaces by singularity type? Our focus will be on cyclic quotient singularities, with an emphasis on discussing concrete visual examples built out of rational, K3, and elliptic surfaces. |
Tuesday, April 27, 2021 Start: 11:00 AM Location: Online via Zoom (Please email the organizers if interested.) |
Jayan Mukherjee University of Kansas | Deformations of Galois Canonical Covers and Applications to Moduli of Surfaces with K^2=4p_g-8. | In this talk we will deal with quadruple Galois canonical covers of surfaces of minimal degree. These surfaces satisfy K_X^2 = 4p_g(X)-8. The earlier case of double covers with K_X^2 = 2p_g(X)-4 was studied by Horikawa . This talk will concentrate on the irregular case. The classification of these surfaces by Gallego-Purnaprajna fall into four irreducible families and their work show that these surfaces behave like general surfaces of general type from various geometric perspectives. We show that except for one family, the general deformation of the canonical morphism is a morphism of degree two onto its image whose normalization is a ruled surface of appropriate genus. We further show that the canonical morphism of a general surface for each of these four families cannot be deformed into a finite birational morphism. As a consequence of our results, we show the existence of infinitely many irreducible uniruled components of the Gieseker moduli space containing surfaces with K_X^2 = 4p_g(X)-8, whose general element is a canonical double cover of a non-normal surface whose normalization is an elliptic ruled surface with invariant e = 0. A general surface of each of these moduli components is unobstructed although H^2(T_X) does not vanish. Our results are in sharp contrast with Horikawa's results on deformations of surfaces of general type with K_X^2 = 2p_g - 4. This is joint work with Purnaprajna Bangere, F.J. Gallego and D. Raychaudhury. |
Tuesday, May 4, 2021 Start: 11:00 AM Location: Online via Zoom (Please email the organizers if interested.) |
Camilla Felisetti Università di Trento | P=W conjectures for character varieties with a symplectic resolution. | Character varieties parametrise representations of the fundamental group of a curve. In general these moduli spaces are singular, therefore it is customary to slightly change the moduli problem and consider smooth analogues, called twisted character varieties. In this setting, the P=W conjecture by de Cataldo, Hausel, and Migliorini suggests a surprising connection between the topology of Hitchin systems and Hodge theory of character varieties. In a joint work with M. Mauri we establish (and in some cases formulate) analogous P=W phenomena in the singular case . In particular we show that the P=W conjecture holds for character varieties which admit a symplectic resolution, namely in genus 1 and arbitrary rank and in genus 2 and rank 2. In the talk I will first mention basic notions of non abelian Hodge theory and introduce the P=W conjecture for smooth moduli spaces; then I will explain how to extend these phenomenas to the singular case, showing the proof our results in a specific example. |
Tuesday, May 11, 2021 Start: 11:00 AM Location: Online via Zoom (Please email the organizers if interested.) |
Christopher Manon University of Kentucky | When is a (projectivized) toric vector bundle a Mori dream space? | Like toric varieties, toric vector bundles are a rich class of varieties which admit a combinatorial description. Following the classification due to Klyachko, a toric vector bundle is captured by a subspace arrangement decorated by toric data. This makes toric vector bundles an accessible test-bed for concepts from algebraic geometry. Along these lines, Hering, Payne, and Mustata asked if the projectivization of a toric vector bundle is always a Mori dream space. Süß and Hausen, and Gonzalez showed that the answer is "yes" for tangent bundles of smooth, projective toric varieties, and rank 2 vector bundles, respectively. Then Hering, Payne, Gonzalez, and Süß showed the answer in general must be "no" by constructing an elegant relationship between toric vector bundles and various blow-ups of projective space, in particular the blow-ups of general arrangements of points studied by Castravet, Tevelev and Mukai. In this talk I'll review some of these results, and then show a new description of toric vector bundles by tropical information. This description allows us to characterize the Mori dream space property in terms of tropical and algebraic data. I'll describe new examples and non-examples, and pose some questions. This is joint work with Kiumars Kaveh. |
Tuesday, May 25, 2021 Start: 11:00 AM Location: Online via Zoom (Please email the organizers if interested.) |
Noble Williamson UC Riverside | The search for effective divisor classes of M_{0,n}. | The Deligne-Mumford compactification of the moduli space of classes of genus zero smooth projective curves with n marked points is a very active object of study in algebraic geometry. One reason for this is that it has some interesting combinatorial structure, that provides abundant tools for its study. A fundamental question about it that remains unsolved is the structure of its cone of numerical equivalence classes of effective divisors. In particular, we want to know which effective divisors generate all others in the cone. We know that the divisors in the boundary are not enough so in this talk we will discuss some clever combinatorial strategies people have used to find non-boundary generating elements of the effective cone. |
Tuesday, June 1, 2021 Start: 11:00 AM Location: Online via Zoom (Please email the organizers if interested.) |
| Talk rescheduled. |
Date | Speaker | Title | Abstract |
Tuesday, January 19, 2021 Start: 11:00 AM Location: Online via Zoom (Please email the organizers if interested.) |
Luca Schaffler KTH Royal Institute of Technology | Compactifications of moduli of points and lines in the projective plane. | Projective duality identifies the moduli space B_{n} parametrizing configurations of n general points in the projective plane with X(3,n), parametrizing configurations of n general lines in the dual projective plane. When considering degenerations of such objects, it is interesting to compare different compactifications of the above moduli spaces. In this work, we consider Gerritzen-Piwek's compactification of B_{n} and Kapranov's Chow quotient compactification of X(3,n), and we show that they have isomorphic normalizations. We also construct an alternative compactification parametrizing all possible n-pointed central fibers of Mustafin joins associated to one-parameter degenerations of n points in the projective plane, which was proposed by Gerritzen and Piwek. We fully describe this alternative compactification for n=5,6. This is joint work with Jenia Tevelev. |
Tuesday, January 26, 2021 Start: 11:00 AM Location: Online via Zoom (Please email the organizers if interested.) |
Changho Han University of Georgia | Compact Moduli of lattice polarized K3 surfaces with nonsymplectic cyclic action of order 3. | Observe that any construction of "meaningful" compactification of moduli spaces of objects involve enlarging the class of objects in consideration. For example, Deligne and Mumford introduced the notion of stable curves in order to compactify the moduli of smooth curves of genus g, and Satake used the periods from Hodge theory to compactify the same moduli space. After a brief review of the elliptic curve case (how those notions are the same), I will generalize into looking at various compactifications of Kondo's moduli space of lattice polarized K3 surfaces (which are of degree 6) with nonsymplectic Z/3Z group action; this involves periods and genus 4 curves by Kondo's birational period map in 2002. Then, I will extend Kondo's birational map to describe birational relations between different compactifications by using the slc compactifications (also known as KSBA compactifications) of moduli of surface pairs. The main advantage of this approach is that we obtain an explicit classification of degenerate K3 surfaces, which is used to find geometric meaning of points parametrized by Hodge-theoretic compactifications. This comes from joint works (in progress) with Valery Alexeev, Anand Deopurkar, and Philip Engel. |
Tuesday, February 2, 2021 Start: 11:00 AM Location: Online via Zoom (Please email the organizers if interested.) |
Dustin Ross San Francisco State University |
Putting the volume back in volume polynomials. | It is a strange and wonderful fact that Chow rings of matroids behave in many ways similarly to Chow rings of smooth projective varieties. Because of this, the Chow ring of a matroid is determined by a homogeneous polynomial called its volume polynomial, whose coefficients record the degrees of all possible top products of divisors. The use of the word "volume" is motivated by the fact that the volume polynomial of a smooth projective toric variety actually measures the volumes of certain polytopes associated to the variety. In the matroid setting, on the other hand, no such polytopes exist, and the goal of our work was to find more general polyhedral objects whose volume is measured by the volume polynomials of matroids. In this talk, I will motivate and describe these polyhedral objects. Parts of this work are joint with Jeshu Dastidar and Anastasia Nathanson. |
Tuesday, February 9, 2021 Start: 11:00 AM Location: Online via Zoom (Please email the organizers if interested.) |
Nathan Ilten Simon Fraser University |
Fano schemes for complete intersections in toric varieties | The study of the set of lines contained in a fixed hypersurface is classical: Cayley and Salmon showed in 1849 that a smooth cubic surface contains 27 lines, and Schubert showed in 1879 that a generic quintic threefold contains 2875 lines. More generally, the set of k-dimensional linear spaces contained in a fixed projective variety X itself is called the k-th Fano scheme of X. These Fano schemes have been studied extensively when X is a general hypersurface or complete intersection in projective space. In this talk, I will report on work with Tyler Kelly in which we study Fano schemes for hypersurfaces and complete intersections in projective toric varieties. In particular, I'll give criteria for the Fano schemes of generic complete intersections in a projective toric variety to be non-empty and of "expected dimension". Combined with some intersection theory, this can be used for enumerative problems, for example, to show that a general degree (3,3)-hypersurface in the Segre embedding of P^2×P^2 contains exactly 378 lines. |
Tuesday, Febraury 16, 2021 Start: 11:00 AM Location: Online via Zoom (Please email the organizers if interested.) |
Brian Collier UC Riverside |
Global Slodowy slices for moduli spaces of Higgs bundles and holomorphic connections. Part 1. | In this first talk I will introduce the moduli space of Higgs bundles on a projective curve and describe some of its properties and structures. I will then explain how the nonabelian Hodge correspondence topologically identifies the Higgs bundle moduli space and the moduli space of local systems and leads to a hyper-Kahler structure on the moduli space. Finally, we will describe how certain affine holomorphic Lagrangian subvarieties of these spaces are related to hyperbolic structures and complex projective structures on the topological surface. |
Tuesday, February 23, 2021 Start: 11:00 AM Location: Online via Zoom (Please email the organizers if interested.) |
Brian Collier UC Riverside |
Global Slodowy slices for moduli spaces of Higgs bundles and holomorphic connections. Part 2. | In this talk, we will discuss the relationship between the Hitchin section in the Higgs bundle moduli space and Beilinson-Drinfeld's parameterizations of space of opers. This generalizes our previous discussion on hyperbolic and projective structures. In these theories, a key role is played by a principal nilpotent and the Borel subgroup. In the remaining time, we will discuss joint work with Andrew Sanders which generalizes this story to arbitrary parabolics. Time permitting, we will discuss how the objects we parameterize are related to higher Teichmüller theory and also mention some open problems. |
Tuesday, March 9, 2021 Start: 11:00 AM Location: Online via Zoom (Please email the organizers if interested.) |
Matthew Satriano University of Waterloo |
New types of heights with connections to the Batyrev-Manin and Malle Conjectures | The Batyrev-Manin conjecture gives a prediction for the asymptotic growth rate of rational points on varieties over number fields when we order the points by height. The Malle conjecture predicts the asymptotic growth rate for number fields of degree d when they are ordered by discriminant. The two conjectures have the same form and it is natural to ask if they are, in fact, one and the same. We develop a theory of point counts on stacks and give a conjecture for their growth rate which specializes to the two aforementioned conjectures. This is joint work with Jordan Ellenberg and David Zureick-Brown. No prior knowledge of stacks will be assumed for this talk. |
Date | Speaker | Title | Abstract |
Tuesday, October 13, 2020 Start: 11:00 AM Location: Online via Zoom (Please email the organizers if interested.) |
Patricio Gallardo UC Riverside | On wonderful blow-ups. | Techniques for constructing good compactifications of an open set is one of the main problems within Algebraic Geometry. In this talk, I will describe a tool known as Wonderful Compactifications due to Li-Li which generalizes earlier work by De Concini-Procesi on hyperplane arrangements. Applications to moduli problems will be described as well. |
Tuesday, October 20, 2020 Start: 11:00 AM Location: Online via Zoom (Please email the organizers if interested.) |
Patricio Gallardo UC Riverside | On wonderful blow-ups (second part). | We will continue describing the theory of wonderful blow-ups. A particular focus is given to applications within moduli theory as well as open problems. |
Tuesday, October 27, 2020 Start: 11:00 AM Location: Online via Zoom (Please email the organizers if interested.) |
Benjamin Schmidt Leibniz Universität Hannover | A curious moduli space on cubic threefolds. | The intermediate Jacobian J(X) of a cubic threefold X was introduced by Clemens and Griffiths in 1972 to prove irrationality of cubic threefolds. It is an abelian variety that can be thought of as parametrizing degree zero cycles in dimension one up to rational equivalence. In this talk we will concentrate on its theta divisor ϴ. Clemens and Griffiths proved the so-called Torelli theorem for cubic threefolds that says that the pair (J(X),ϴ) determines the cubic threefold. Shortly after, Mumford pointed out that X can be recovered just from the singularities of the theta divisor. In fact, it has a unique singularity whose tangent cone is the affine cone of the cubic X. Blowing the singularity up yields a resolution of singularities. We will construct this resolution as a moduli space of rank three vector bundles. This allows us to recover the so-called derived Torelli theorem. It roughly says that a certain triangulated subcategory (called the Kuznetsov component of X) of the bounded derived category of coherent sheaves determines X. |
Tuesday, November 3, 2020 Start: 11:00 AM Location: Online via Zoom (Please email the organizers if interested.) |
US Election Day | Election Day. | No seminar meeting on US Election Day. |
Tuesday, November 10, 2020 Start: 11:00 AM Location: Online via Zoom (Please email the organizers if interested.) |
Olivia Dumitrescu University of North Carolina at Chapel Hill | Lagrangian correspondence between Hitchin and de Rham moduli spaces. | In 2008 Simpson conjectures the existence of a holomorphic Lagrangian foliation in the de Rham moduli space of holomorphic G-connections for a complex reductive group G. The purpose of the talk is to establish the existence of a holomorphic Lagrangian foliation in the de Rham moduli space of holomorphic SL_2(C)-connections defined on a smooth connected projective curve C of genus at least 2. The conjectural holomorphic Lagrangian foliation does not seem to constitute a holomorphic Lagrangian fibration. I will present an algebraic geometry description of the Lagrangian correspondence of conformal limits, based on the work of Simpson. This talk is based on joint work with Jennifer Brown and Motohico Mulase. |
Tuesday, November 17, 2020 Start: 11:00 AM Location: Online via Zoom (Please email the organizers if interested.) |
Dagan Karp Harvey Mudd College | The Chow ring of heavy/light Hassett spaces via tropical geometry. | Hassett spaces in genus 0 are moduli spaces of weighted pointed stable rational curves; they are important in the minimal model program and enumerative geometry. We compute the Chow ring of heavy/light Hassett spaces. The computation involves intersection theory on the toric variety corresponding to a graphic matroid, and rests upon the work of Cavalieri-Hampe-Markwig-Ranganathan. This is joint work with Siddarth Kannan and Shiyue Li. Slides |
Tuesday, November 24, 2020 Start: 11:00 AM Location: Online via Zoom (Please email the organizers if interested.) |
Antonio Laface. University of Concepcion | Blown-up toric surfaces with non-polyhedral effective cone. | I will discuss examples of projective toric surfaces whose blow-up at a general point has a non-polyhedral pseudoeffective cone, both in characteristic 0 and in positive characteristic. As a consequence, the pseudo-effective cone of the Grothendieck-Knudsen moduli space M_{0,n} is not polyhedral for n ≥ 10 in characteristic 0 and for an infinite set of primes of positive density in positive characteristic. |
Tuesday, December 1, 2020 Start: 11:00 AM Location: Online via Zoom (Please email the organizers if interested.) |
Javier Gonzalez-Anaya UC Riverside | Negative curves in blowups of weighted projective planes. | Blowups of toric varieties at general points have played a central role in many recent developments concerning the birational geometry of some moduli spaces. As part of this ongoing program, we'll discuss what is known about the Mori dream space property for blowups of weighted projective planes at a general point. By a result of Cutkosky, such a variety is a Mori dream space if and only if it contains two non-exceptional irreducible curves disjoint from each other; one of them having non-positive self-intersection. Such a curve a is called a “negative curve”. Negative curves largely govern the Mori dream space property for these varieties. In this talk I will survey what is currently known about their existence, how they "interact" with each other and how these interactions inform us about the Mori dream space property in many cases. |
Tuesday, December 8, 2020 Start: 11:00 AM Location: Online via Zoom (Please email the organizers if interested.) |
Bernt Ivar Utstøl Nødland Norwegian Defence Research Establishment | Cox rings of projectivized toric vector bundles. | A toric vector bundle is a torus equivariant vector bundle on a toric variety. To a toric vector bundle one can associate a collection of lattice polytopes called the parliament of polytopes. We show how we can use these polytopes to give a description of the Cox ring of a projectivized toric vector bundle. |