April 23, 2022
Talk 1: Carl Lian (HU Berlin) (Video)
Title: Tevelev degrees of hypersurfaces
Abstract: We consider the following problem: let (C,p_1,…,p_n) be a fixed general pointed curve of genus g, let X be a smooth hypersurface, and let x_1,…,x_n be general points on X. Then, how many degree d morphisms f:C->X are there for which f(p_i)=x_i? This problem has been (largely, but not completely) solved „virtually“ in Gromov-Witten theory by Buch-Pandharipande and Cela. The virtual counts are expected to be enumerative if d is sufficiently large, but this is only known for hypersurfaces of very low degree (joint with Pandharipande).
I will describe a more recent elementary approach to the problem via projective geometry, which recovers the virtual counts. The main difficulty is to analyze the transversality of the intersection in question, analogously to the prior investigation with Pandharipande. This leads to questions on bounding excess dimensions of certain families of singular curves on hypersurfaces which remain open.
Talk 2: Eric Jovinelly (Notre Dame) (Video)
Title: Extreme Divisors on M_{0,7} and Differences over Characteristic 2
Abstract: The cone of effective divisors controls the rational maps from a variety. We study this important object for M_{0,n}, the moduli space of stable rational curves with n markings. Fulton once conjectured the effective cones for each n would follow a certain combinatorial pattern. However, this pattern holds true only for n < 6. Despite many subsequent attempts to describe the effective cones for all n, we still lack even a conjectural description. We study the simplest open case, n=7, and identify the first known difference between characteristic 0 and characteristic p. Although a full description of the effective cone for n=7 remains open, our methods allowed us to compute the entire effective cones of spaces associated with other stability conditions.
March 26, 2022
Wern Yeong (Notre Dame) (Video)
Title: Algebraic hyperbolicity of very general hypersurfaces in products of projective spaces
Abstract: A complex algebraic variety is said to be hyperbolic if it contains no entire curves, which are non-constant holomorphic images of the complex line. Demailly introduced algebraic hyperbolicity as an algebraic version of this property, and it has since been well-studied as a means for understanding Kobayashi’s conjecture, which says that a generic hypersurface in projective space is hyperbolic whenever its degree is large enough. In this talk, we study the algebraic hyperbolicity of very general hypersurfaces of high bi-degrees in Pm x Pn and completely classify them by their bi-degrees, except for a few cases in P3 x P1. We present three techniques to do that, which build on past work by Ein, Voisin, Pacienza, Coskun and Riedl, and others. As another application of these techniques, we improve the known result that very general hypersurfaces in Pn of degree at least 2n − 2 are algebraically hyperbolic when n is at least 6 to when n is at least 5, leaving n = 4 as the only open case.
February 26, 2022
Title: Moduli spaces of linear maps with marked points
Abstract: Moduli spaces of degree d dynamical systems on projective space are fundamental in algebraic dynamics. When the degree d is at least 2, these moduli spaces can be defined via geometric invariant theory (GIT). But when d = 1, there is a fundamental problem: there are no GIT stable linear maps. Inspired by the case of genus 0 curves, we show how to recover a nice moduli space by including marked points. We construct the moduli space of linear maps with marked points, prove its rationality, and show that GIT stability is characterized by subtle dynamical conditions on the marked map. The proof is a combinatorial analysis of polytopes generated by root vectors of the A_N lattice from Lie theory.
Title: Open/closed correspondence via relative/local correspondence
Abstract: We discuss a mathematical approach to the open/closed correspondence proposed by Mayr, which is a correspondence between the disk invariants of toric Calabi-Yau threefolds and genus-zero closed Gromov-Witten invariants of toric Calabi-Yau fourfolds. We establish the correspondence in two steps: First, a correspondence between the disk invariants and the genus-zero maximally-tangent relative Gromov-Witten invariants of relative Calabi-Yau threefolds, which follows from the topological vertex (Li-Liu-Liu-Zhou, Fang-Liu). Second, a correspondence between the maximally-tangent relative invariants and the closed invariants, which can be viewed as an instantiation of the log-local principle of van Garrel-Graber-Ruddat in the non-compact setting. Our correspondences are based on localization. We also discuss generalizations and implications of our correspondences. Joint work with Chiu-Chu Melissa Liu.
January 29, 2022
Title: Stable cohomology of the moduli space of trigonal curves
Abstract: The rational cohomology of the moduli space T_g of trigonal curves of genus g has been computed by Looijenga for g=3, by Tommasi for g=4 and by myself for g=5. In this talk I will present the rational cohomology of T_g for higher genera. Specifically, we prove that it is independent of i for g>4i+3 and that it coincides with the tautological ring in this range. This will be done by studying the embedding of trigonal curves in Hirzebruch surfaces and using Gorinov-Vassiliev's method.
Title: Geometry of q-bic Hypersurfaces
Abstract: Let’s count: 1, 2, q+1. The eponymous objects are special projective hypersurfaces of degree q+1, where q is a power of the positive ground field characteristic. In this talk, I would like to sketch an analogy between the geometry of q-bic hypersurfaces and that of quadric and cubic hypersurfaces. For instance, the moduli spaces of linear spaces in q-bics are smooth and themselves have rich geometry. In the case of q-bic threefolds, I will describe an analogue of result of Clemens and Griffiths, which relates the intermediate Jacobian of the q-bic with the Albanese of its surface of lines.
November 20, 2021
Talk 1: Shengxuan Liu (Warwick) (Video)
Title: Stability condition on Calabi-Yau threefold of complete intersection of quadratic and quartic hypersurfaces
Abstract: In this talk, I will first introduce the background of Bridgeland stability condition. Then I will mention some existence result of Bridgeland stability. Next I will prove the Bogomolov-Gieseker type inequality of X_(2,4), Calabi-Yau threefold of complete intersection of quadratic and quartic hypersufaces, by proving the Clifford type inequality of the curve X_(2,2,2,4). This will provide the existence of Bridgeland stability condition of X_(2,4).
Talk 2: Heather Lee (Video)
Title: Counting special Lagrangian classes and semistable Mukai vectors for K3 surfaces
Abstract: Motivated by the study of the growth rate of the number of geodesics in flat surfaces with bounded lengths, we study generalizations of such problems for K3 surfaces. In one generalization, we give a result regarding the upper bound on the asymptotics of the number of classes of irreducible special Lagrangians in K3 surfaces with bounded period integrals. In another generalization, we give the exact leading term in the asymptotics of the number of Mukai vectors of semistable coherent sheaves on algebraic K3 surfaces with bounded central charges, with respect to generic Bridgeland stability conditions. (I will provide all the necessary background for the terminologies that appear here during the talk, so it's not necessary for the audience to know them beforehand.) This talk is based on joint work with Jayadev Athreya and Yu-Wei Fan.
September 18, 2021
Talk 1: Dori Bejleri (Harvard) (Video)
Title: Wall crossing for moduli of stable log varieties
Abstract: Stable log varieties or stable pairs (X,D) are the higher dimensional generalization of pointed stable curves. They form proper moduli spaces which compactify the moduli space of normal crossings, or more generally klt, pairs. These stable pairs compactifications depend on a choice of parameters, namely the coefficients of the boundary divisor D. In this talk, after introducing the theory of stable log varieties, I will explain the wall-crossing behavior that governs how these compactifications change as one varies the coefficients. I will also discuss some examples and applications. This is joint work with Ascher, Inchiostro, and Patakfalvi.
Talk 2: Yixian Wu (UT Austin) (Video)
Title: Splitting of Gromov-Witten Invariants with Toric Gluing Strata
Abstract: For the past decades, relative Gromow-Witten theory and the degeneration formula have been proved to be an important technique in computing Gromov-Witten invariants. The recent development of logarithmic and punctured Gromov-Witten theory of Abramovich, Chen, Gross and Siebert generalizes the theories to normal crossing varieties. The natural next step is to obtain a degeneration formula under the normal crossing degeneration. In this talk, I will present a formula relating the Gromov-Witten invariants of general fibers to the strata of invariants of components of the central fiber, with the assumption that the gluing happens at toric varieties. I will explain how tropical geometry naturally arises and provides the key tool for the formula.
May 22, 2021
Talk 1: Rohini Ramadas (Brown University)
Title: Special loci in the moduli space of self-maps of projective space
Abstract: A self-map of P^n is called post critically finite (PCF) if its critical hypersurface is pre-periodic. I’ll give a survey of many known results and some conjectures having to do with the locus of PCF maps in the moduli space of self-maps of P^1. I’ll then present a result, joint with Patrick Ingram and Joseph H. Silverman, that suggests that for n≥2, PCF maps are comparatively scarce in the space of self-maps of P^n. I’ll also mention joint work with Rob Silversmith, and work-in-progress with Xavier Buff and Sarah Koch, on loci of “almost PCF” maps of P^1.
Talk 2: Rosa Schwarz (Leiden University) (Video)
Title: The universal and log double ramification cycle
Abstract: The double ramification cycle is a class most commonly studied on the moduli space of marked curves. In joint work with Y. Bae, D. Holmes, R. Pandharipande, and J. Schmitt, we define the universal double ramification cycle in the operational Chow group of the Picard stack (of Jacobian). Even though we name it the universal double ramification cycle, I would like to define this cycle and then explain why this is not the final most natural DR-cycle to consider. For example, it does not satisfy some basic properties about intersecting these cycles (the double double ramification cycle) that intuitively should hold. In fact, we need to consider certain log-blowups of the Picard stack as well. This results in a log DR-cycle on a log Chow ring, which does satisfy these nice intersection properties. Moreover, we can ask and answer questions such as whether this DR-cycle is log tautological. This talk is based on recent joint work with D. Holmes. (Some of this talk wil be closely related to what Sam Molcho discussed in his talk in this seminar, but the general approach is quite different).
April 24, 2021
Title: Topology of tropical moduli spaces of weighted stable curves in higher genus
Abstract: 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, 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. I will also discuss some work in progress on a geometric group theoretic approach to the simple connectivity of these spaces. This is joint work with Siddarth Kannan, Stefano Serpente and Claudia Yun.
Title: The Chow rings of the moduli space of curves of genus 7, 8, and 9
Abstract: The rational Chow ring of the moduli space of smooth curves is known when the genus is at most 6 by work of Mumford (g=2), Faber (g=3, 4), Izadi (g=5), and Penev-Vakil (g=6). In each case, it is generated by the tautological classes. On the other hand, van Zelm has shown that the bielliptic locus is not tautological when g=12. In recent joint work with Hannah Larson, we show that the Chow rings of M_7, M_8, and M_9 are generated by tautological classes, which determines the Chow ring by work of Faber. I will explain an overview of the proof with an emphasis on the special geometry of curves of low genus and low gonality.
March 27, 2021
Talk 1: Rob Silversmith (Northeastern) (Video)
Title: Stratifications of Hilbert schemes from tropical geometry
Abstract: One may associate, to any homogeneous ideal I in a polynomial ring, a combinatorial shadow called the tropicalization of I. In any Hilbert scheme, one may consider the set of ideals with a given tropicalization; these are the strata of the “tropical stratification" of the Hilbert scheme. I will discuss some of the many questions one can ask about tropicalizations of ideals, and how they are related to some classical questions in combinatorial algebraic geometry, such as the classification of torus orbits on Hilbert schemes of points in C^2. Some unexpected combinatorial objects appear: e.g. when studying tropicalizations of subschemes of P^1, one is led to Schur polynomials and binary necklaces. This talk includes joint work with Diane Maclagan.
Talk 2: Sam Molcho (ETH) (Video)
Title: The logarithmic tautological ring
Abstract: Let (X,D) be a pair consisting of a smooth variety X with a normal crossings divisor D. In this talk, I will discuss the construction of a subring of the Chow ring of X, called the logarithmic tautological ring, generated by certain "tautological" classes obtained from the strata of D. I will explain the basic structure of the logarithmic tautological ring: its behavior under blowups, its relation to combinatorics, and some methods to compute it. I will conclude by relating the logarithmic tautological ring of the moduli space of curves with the double ramification cycle, and explain how the structure of the logarithmic tautological ring implies that the double ramification cycle is a product of divisors in a blowup of \bar{M}_{g,n}.
February 27, 2021
Talk 1: Irene Schwarz (Humboldt University of Berlin) (Video)
Title: On the Kodaira dimension of the moduli space of hyperelliptic curves with marked points
Abstract: It is known that the moduli space Hg,n of genus g stable hyperelliptic curves with n marked points is uniruled for n ≤ 4g + 5. We consider the complementary case and show that Hg,n has non-negative Kodaira dimension for n = 4g+6 and is of general type for n ≥ 4g+7. Important parts of our proof are the calculation of the canonical divisor and establishing that the singularities of Hg,n do not impose adjunction conditions.
Talk 2: Man-Wai Mandy Cheung (Harvard University) (Video)
Title: Compactifications of cluster varieties and convexity
Abstract: Cluster varieties are log Calabi-Yau varieties which are unions of algebraic tori glued by birational "mutation" maps. They can be seen as a generalization of the toric varieties. In toric geometry, projective toric varieties can be described by polytopes. We will see how to generalize the polytope construction to cluster convexity which satisfies piecewise linear structure. As an application, we will see the non-integral vertex in the Newton Okounkov body of Grassmannian comes from broken line convexity. We will also see links to the symplectic geometry and application to mirror symmetry. The talk will be based on a series of joint works with Bossinger, Lin, Magee, Najera-Chavez, and Vianna.
January 23, 2021
Talk 1: Fatemeh Rezaee (Loughborough University)
Title: Minimal Model Program via wall-crossing in higher dimensions?
Abstract: In this talk, I will explain a new wall-crossing phenomenon on P^3 that induces non-Q-factorial singularities and thus cannot be understood as an operation in the Minimal Model Program of the moduli space, unlike the case for many surfaces. I will start by giving a review of Bridgeland stability conditions on derived categories.
Talk 2: Michel van Garrel (University of Birmingham) (Video)
Title: Stable maps to Looijenga pairs
Abstract: Start with a rational surface Y admitting a decomposition of its anticanonical divisor into at least 2 smooth nef components. We associate 5 curve counting theories to this Looijenga pair: 1) all genus stable log maps with maximal tangency to each boundary component; 2) genus 0 stable maps to the local Calabi-Yau surface obtained by twisting Y by the sum of the line bundles dual to the components of the boundary; 3) the all genus open Gromov-Witten theory of a toric Calabi-Yau threefold associated to the Looijenga pair; 4) the Donaldson-Thomas theory of a symmetric quiver specified by the Looijenga pair and 5) BPS invariants associated to the various curve counting theories. In this joint work with Pierrick Bousseau and Andrea Brini, we provide closed-form solutions to essentially all of the associated invariants and show that the theories are equivalent. I will start by describing the geometric transitions from one geometry to the other, then give an overview of the curve counting theories and their relations. I will end by describing how the scattering diagrams of Gross and Siebert are a natural place to count stable log maps.
November 21, 2020
Talk 1: Sebastian Bozlee (Tufts University) (Video)
Title: Contractions of logarithmic curves and alternate compactifications of the space of pointed elliptic curves
Abstract: There are many ways to construct proper moduli spaces of pointed curves of genus 1, among them the spaces of Deligne-Mumford stable curves, pseudostable curves, and m-stable curves. These spaces are birational to each other, and earlier work by Ranganathan, Santos-Parker, and Wise has shown that logarithmic geometry gives us a nice system for resolving the rational maps between them: first one performs some blowups, then one applies a contraction to a universal family. In my thesis, I construct a contraction map for more general families of log curves. Systematic exploration of the possible contractions of universal families (joint with Bob Kuo and Adrian Neff) uncovers new semistable modular compactifications of the space of pointed elliptic curves of genus 1.
We will start with a description of the moduli spaces, discuss some basics of log geometry, then describe the contraction construction. Time permitting, we will sketch the process of finding contractions of universal families permitted by the construction.
Talk 2: Francesca Carocci (École Polytechnique Fédérale de Lausanne) (Video)
Title: A modular smooth compactification of genus 2 curves in projective spaces
Abstract: Moduli spaces of stable maps in genus bigger than zero include many components of different dimensions meeting each other in complicated ways, and the closure of the smooth locus is difficult to describe modularly.
After the work of Li--Vakil--Zinger and Ranganathan--Santos-Parker--Wise in genus one, we know that points in the boundary of the main component correspond to maps that admit a factorisation through some curve with Gorenstein singularities on which the map is less degenerate.
The question becomes how to construct such a universal family of Gorenstein curves to then single out the (resolution) of the main component of maps imposing the factorization property. In joint work with L. Battistella, we construct one such family in genus two over a logarithmic modification of the space of admissible covers.
October 31, 2020
Talk 1: Frederik Benirschke (Stony Brook University) (Video)
Title: Compactification of linear subvarieties
Abstract: The moduli space of differential forms on Riemann surfaces, also known as stratum of differentials, has natural coordinates given by the periods of the differential. A very special class of subvarieties of strata is given by linear subvarieties. These are algebraic subvarieties of strata which are given locally by linear equations among the periods. Interesting examples of linear varieties arise from both algebraic geometry as well as Teichmüller theory. Using the recent compactification of strata developed by Bainbridge-Chen-Gendron-Grushevsky-Möller we construct an algebraic compactification of linear subvarieties and study its properties. Our main result is that the boundary of a linear subvariety is again given by linear equations among periods. Time permitting, we show how our results can be used to study Hurwitz spaces.
Talk 2: Hülya Argüz (Université de Versailles, Paris-Saclay) (Video)
Title: Enumerating punctured log Gromov-Witten invariants from wall-crossing
Abstract: Log Gromov-Witten theory developed by Abramovich-Chen and Gross-Siebert concerns counts of stable maps with prescribed tangency conditions relative to a (not necessarily smooth) divisor. An extension of log Gromov-Witten theory to the case where one allows negative tangencies is provided by punctured log Gromov-Witten theory of Abramovich-Chen-Gross-Siebert. In this talk we describe an algorithmic method to compute punctured log Gromov-Witten invariants of log Calabi-Yau varieties obtained from blow-ups of toric varieties along hypersurfaces on the toric boundary. This method uses tropical geometry and wall-crossing computations. This is joint work with Mark Gross.
September 26, 2020
Talk 1: Dennis Tseng (MIT) (Video)
Title: Algebraic Geometry and the Log-Concavity of Matroid Invariants
Abstract: In their celebrated paper, Adiprasito, Huh, and Katz showed the coefficients of the characteristic polynomial of any matroid form a log-concave sequence. In an effort to interest algebraic geometers, we introduce the geometric side of the story, which applies when the matroid is representable. In this story, we will encounter familiar spaces, like Grassmannians and toric varieties. We will also see variations on this geometric setup, leading to joint work with Andrew Berget and Hunter Spink, and preliminary work with the aforementioned authors and Christopher Eur.
Talk 2: Hannah Larson (Stanford University) (Video)
Title: Brill--Noether theory over the Hurwitz space
Abstract: Let C be a curve of genus g. A fundamental problem in the theory of algebraic curves is to understand maps of C to projective space of dimension r of degree d. When the curve C is general, the moduli space of such maps is well-understood by the main theorems of Brill--Noether theory. However, in nature, curves C are often encountered already equipped with a map to some projective space, which may force them to be special in moduli. The simplest case is when C is general among curves of fixed gonality. Despite much study over the past three decades, a similarly complete picture has proved elusive in this case. In this talk, I will discuss recent joint work with Eric Larson and Isabel Vogt that completes such a picture, by proving analogs of all of the main theorems of Brill--Noether theory in this setting.