Harvard/MIT Algebraic Geometry Seminar

Spring 2019
Tuesdays at 3 pm

The Harvard/MIT Algebraic Geometry Seminar will alternate between MIT (2-147) and Harvard (Science Center 507).

• Feb 52018 SC 507 Harvard Mihai Fulger, University of Connecticut
  Seshadri constants for bundles abstract±    For line bundles, Seshadri constants are classical local positivity invariants at a point. They detect ampleness in several ways, and they measure jet separation at the point in an asymptotic sense. A generalization of Seshadri constants to higher rank was first studied explicitly by Hacon. In our work, we extend the classical rank one results. We also give three applications. First, we offer a Seshadri characterization of projective space as the only Fano manifold whose tangent bundle has positive Seshadri constant for at least one point. We conjecture that the Fano assumption can be removed, and offer the case of surfaces as evidence. Second, we make partial progress towards a conjecture on the shape of the nef cone of the self product of a very general curve of high genus. Third, we observe that Seshadri constants control jet separation for direct images of pluricanonical sheaves. This is all in joint work with Takumi Murayama.
• Feb 122019 SC 507 Harvard Dan Abramovich, Brown University
  Moduli technique in resolution of singularities abstract±    Semistable reduction, a form of resolution of singularities of families, is often the first step in constructing compactified moduli spaces, and can be used to discover their properties. I will describe work-in-progress with Michael Temkin and Jaroslaw Wlodarczyk in which we prove functorial semistable reduction for families of varieties in characteristic 0, refining work with Karu from 2000. Techniques developed for moduli spaces enter in unexpected ways.
• Feb 192018 2-147 MIT Maksym Fedorchuk, Boston College
  Standard models of low degree del Pezzo fibrations abstract±    A del Pezzo fibration is one of the natural outputs of the Minimal Model Program for threefolds. At the same time, geometry of an arbitrary del Pezzo fibration can be unsatisfying due to the presence of non-integral fibers and terminal singularities of an arbitrarily large index. In 1996, Corti developed a program of constructing `standard models' of del Pezzo fibrations within a fixed birational equivalence class. Standard models enjoy a variety of desired properties, one of which is that all of their fibers are Q-Gorenstein integral del Pezzo surfaces. Corti proved the existence of standard models for del Pezzo fibrations of degree d\geq 2, with the case of d=2 being the most difficult. The case of d=1 remained a conjecture. In 1997, Kollár recast and improved the Corti’s result in degree d=3 using ideas from the Geometric Invariant Theory for cubic surfaces. I will present a generalization of Kollár’s approach in which we develop notions of stability for families of low degree (d\leq 2) del Pezzo fibrations in terms of their Hilbert points (i.e., low degree equations cutting out del Pezzos). A correct choice of stability and a bit of enumerative geometry then leads to (very good) standard models in the sense of Corti. This is a joint work with Hamid Ahmadinezhad and Igor Krylov.
• Feb 262019 SC B10 Harvard Ethan Cotterill, Universidade Federal Fluminense (**Note unusual room**)
  Real inflection points of real linear series on real (hyper)elliptic curves (joint with I. Biswas and C. Garay López) abstract±    According to Plucker's formula, the total inflection of a linear series (L,V) on a complex algebraic curve C is fixed by numerical data, namely the degree of L and the dimension of V. Equipping C and (L,V) with compatible real structures, it is more interesting to ask about the total real inflection of (L,V). The topology of the real inflectionary locus depends in a nontrivial way on the topology of the real locus of C. We study this dependency when C is hyperelliptic and (L,V) is a complete series. We first use a nonarchimedean degeneration to relate the (real) inflection of complete series to the (real) inflection of incomplete series on elliptic curves; we then analyze the real loci of Wronskians along an elliptic curve, and formulate some conjectural quantitative estimates.
• Mar 52019 2-147 MIT Sho Tanimoto, Kumamoto University
  Sections on del Pezzo fibrations over P^1 abstract±    We prove movable bend and break lemma, i.e., a free curve of high degree breaks into the union of two free curves as a stable map for sections on del Pezzo fibrations over P^1. We use this to study some inductive arguments proving the irreducibility of the space of sections. Our approach is inspired by Manin’s Conjecture and also has applications to it. This is joint work with Brian Lehmann.
• ThurMar 7 2-449 MIT Yongbin Ruan, University of Michigan (**Special seminar at 1 PM; Note unusual room/date/time**)
  The structure of higher genus Gromov-Witten theory of quintic 3-folds abstract±    One of biggest and most difficult problems in the subject of Gromov-Witten theory is to compute higher genus Gromov-Witten theory of compact Calabi-Yau 3-fold. There have been a collection of remarkable conjectures from physics for so called 14 one-parameter models, simplest compact Calabi-Yau 3-folds similar to the quintic 3-folds. The backbone of this collection of conjectures are four structural conjectures: (1) Yamaguchi-Yau finite generation; (2) Holomorphic anomaly equation; (3) Orbifold regularity and (4) Conifold gap condition. In the talk, I will present background and our approach to the problem. This is a joint work with F. Janda and S. Guo. Our proof is based on certain localization formula from log GLSM theory developed by Q. Chen, F. Janda and myself.
• ThurMar 7 2-449 MIT Felix Janda, University of Michigan (**Special seminar at 2:30 PM; Note unusual room/date/time**)
  Holomorphic anomaly equations for quintic threefolds abstract±    The generating series F_g of Gromov-Witten invariants of Calabi-Yau threefolds are conjectured to behave like quasi-modular forms. Furthermore, there are holomorphic anomaly equations that control in which way F_g fails to be modular. These conjectures have been used very successfully in the physics literature, for example in the computation of Gromov-Witten invariants of quintics up to genus 51. In my talk I will explain how to use the new technique of log GLSM (under development with Q. Chen and Y. Ruan) to give a proof of these conjectures in the case of quintic threefolds. Based on joint work with S. Guo and Y. Ruan.
• Mar 122019 SC 507 Harvard Jake Levinson, University of Washington
  Boij-Söderberg Theory for Grassmannians abstract±    The Betti table of a graded module over a polynomial ring encodes much of its structure and that of the corresponding sheaf on projective space. In general, it is hard to tell which tables of numbers can arise as Betti tables. An easier problem is to describe such tables up to positive scalar multiple: this is the "cone of Betti tables". The Boij-Söderberg conjectures, proven by Eisenbud-Schreyer, gave a beautiful description of this cone and, as a bonus, a "dual" description of the cone of cohomology tables of sheaves. I will describe some extensions of this theory, joint with Nicolas Ford and Steven Sam, to the setting of GL-equivariant modules over coordinate rings of matrices. Here, the dual theory (in geometry) concerns sheaf cohomology on Grassmannians. The main result is an equivariant analog of the Boij-Söderberg pairing between Betti tables and cohomology tables. This is a bilinear pairing of cones, with output in the cone coming from the "base case" of square matrices, which we also fully characterize.
• Mar 192019 2-449 MIT Luc Illusie, Université Paris-Sud, Talk 1, 3-4 (**Note two seminar talks and unusual room**)
  The de Rham-Witt complex: review and prospects abstract±    I will recall the historical background, some of the main results, and discuss open questions raised by the new approach of Bhatt, Lurie and Mathew.
• Mar 192019 2-449 MIT Luc Illusie, Université Paris-Sud, Talk 2, 4:30-5:30
  Remarks on the cotangent complex and the Nygaard filtration: DI revisited abstract±    I'll revisit decompositions of de Rham complexes in positive characteristic (Deligne-Illusie), by discussing some relations between the cotangent complex, liftings mod $p^2$, and the de Rham-Witt complex.
• Mar 262019 SC 507 Harvard Changho Han, Harvard University
  'Almost K3' stable log surfaces and curves of genus 4 abstract±    The geometry of compact moduli spaces of log surfaces is mysterious in general. Thus, describing an example with its geometric properties is already valuable. To do so, we consider an 'almost K3' stable log surface, which is a pair where the log canonical divisor is positive but very close to 0. We study compactified moduli spaces of such log surfaces, constructed using the techniques of Kollár, Shepherd-Barron, Alexeev, and Hacking. We completely describe such a compactification of the moduli space of (X, D) where X is a quadric surface and D is a canonical genus 4 curve, obtaining a new birational model of the moduli space M_4 of smooth curves of genus 4. This is joint work with Anand Deopurkar.
• Apr 22019 2-147 MIT Tomasz Szemberg, Pedagogical University of Cracow
  Postulation in projective spaces and unexpected hypersurfaces abstract±    Given a subscheme X in projective space, it is a very classical problem to determine the dimension of the vector space of all homogeneous polynomials of some fixed degree d vanishing along X. Even in the simplest case, when X is supported on a finite set of points in the projective plane, a complete answer is not known and it is subject to open conjectures due to Nagata (1959) and Segre-Harbourne-Gimigliano-Hirschowitz (1964-78). Recently Cook II, Harbourne, Migliore and Nagel observed that the situation becomes even more interesting if one replaces the space of all polynomials of certain degree by its carefully chosen subspaces. I will focus on examples exhibiting these new phenomena. This is based on joint work with Bauer, Malara and Szpond.
• Apr 92019 SC 507 Harvard Aaron Pixton, MIT
  Kappa rings and boundary vanishing abstract±    The moduli space of compact type curves of genus g is an open locus inside the moduli space of stable curves consisting of those curves with compact Jacobians. The full intersection theory of this compact type moduli space is not very well understood, but in 2009 Pandharipande determined the structure of the kappa ring, a small subring of its Chow ring. After reviewing this work, I'll explain how to use it to construct a nonzero class that restricts to zero on every boundary divisor of the moduli space. I'll conclude by giving a couple conjectures about a similar class on the moduli space of stable curves.
• Apr 162019 2-147 MIT Junho Peter Whang, MIT
  Geometry and arithmetic on moduli of local systems abstract±    Moduli spaces of local systems on topological surfaces, together with their mapping class group dynamics, are widely studied in geometry. Focusing on the special linear rank two case, we discuss geometric and arithmetic properties of these moduli spaces, such as their log-Calabi-Yau nature and structure of polynomial curves/integral points, emphasizing their relation to elementary results on surfaces obtained by combinatorial and differential geometric methods. Time permitting, we discuss some recent works further pursuing the underlying theme.
• Apr 232019 SC 507 Harvard Kalina Mincheva, Yale University
  Tropical Algebra abstract±    Tropical geometry provides a new set of purely combinatorial tools, which has been used to approach classical problems. In tropical geometry most algebraic computations are done on the classical side - using the algebra of the original variety. The theory developed so far has explored the geometric aspect of tropical varieties as opposed to the underlying (semiring) algebra and there are still many commutative algebra tools and notions without a tropical analogue. In the recent years, there has been a lot of effort dedicated to developing the necessary tools for commutative algebra using different frameworks, among which prime congruences, tropical ideals, tropical schemes. These approaches allows for the exploration of the properties of tropicalized spaces without tying them up to the original varieties and working with geometric structures inherently defined in characteristic one (that is, additively idempotent) semifields. In this talk we explore the relationship between tropical ideals and congruences and what they remember about the geometry of a tropical variety. This is joint work with D. Jo\'o.
• Apr 302019 2-147 MIT Daniel Halpern-Leistner, Cornell University
  Categorical stable envelopes abstract±    I will discuss a categorification of the stable basis in the equivariant cohomology of an algebraic symplectic variety with a torus action, introduced by Maulik and Okounkov in their work on the quantum cohomology of Nakajima quiver varieties. The categorical stable envelopes are certain objects in the equivariant derived category of coherent sheaves characterized by support and weight conditions, and they are part of a more general story involving semiorthogonal decompositions of equivariant derived categories. I will discuss this construction, which also provides the first general construction of stable bases in the equivariant K-theory of algebraic symplectic varieties (although other constructions exist in the case of quiver varieties). This is joint work with Davesh Maulik and Andrei Okounkov.
• May 72019 SC 507 Harvard Igor Krylov, Korea Institute for Advanced Study
  Birational rigidity of low degree del Pezzo fibrations abstract±    A Mori fiber space is called birationally rigid if, roughly speaking, it has only one Mori fiber space structure. I will talk about birational rigidity of singular del Pezzo fibrations. I will explain the essence of the method of proving birational rigidity: Noether-Fano method, also known as method of maximal singularities. I will discuss the differences in applying this to Fano varieties and Mori fiber spaces with a positive dimensional base.
• May 142019 2-147 MIT Daniel Bragg, Berkeley
  Supersingular twistor spaces abstract±    We will describe how the crystalline cohomology of a supersingular K3 surface gives rise to certain one-parameter families of K3 surfaces, which we call supersingular twistor spaces. Our construction relies on the special behavior of p-torsion classes in the Brauer group of a supersingular K3 surface. As an application, we will describe a new proof of Ogus's crystalline Torelli theorem, which we extend to small characteristic.

This seminar is organized by Joe Harris (Harvard), Davesh Maulik (MIT), Brooke Ullery (Harvard). This seminar is supported in part by grants from the NSF. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.