Abstracts

Benjamin Bakker (UGA)

Title: o-minimal GAGA and Applications to Hodge Theory

Abstract: A complex algebraic variety can be naturally considered as a complex analytic space. The analytic category is often more flexible, but for this perspective to be useful in algebraic geometry it is necessary to know which analytic constructions produce algebraic objects. One powerful answer to this question is provided by Serre's celebrated GAGA theorem: it says that for a projective variety the algebraic and analytic categories of coherent sheaves are in fact naturally equivalent. This theorem easily fails for non-proper varieties. In joint work with Y. Brunebarbe and J. Tsimerman, we show that a GAGA theorem holds in the non-proper case if one restricts to analytic structures that are "tame" in a sense made precise by the notion of o-minimality. We will also describe why this result has a number of important applications to Hodge theory.

• Moises Herradon Cueto (University of Wisconsin)

  The local type of difference equations

  Abstract: D-modules allow us to study differential equations through the lens of algebraic geometry. They are widely studied and have been shown to be full of structure. In contrast, the case of difference equations is lacking some of the most basic constructions. We focus on the following question: D-modules have a clear notion of what it means to restrict to a (formal) neighborhood of a point, namely extension of scalars to a power series ring. However, what does it mean to restrict a difference equation to a neighborhood of a point? I will give an answer which encompasses the intuitive notions of a "zero" and a "pole" of a difference equation, but further it is consistent in two more ways. First of all, we can show that restricting a difference equation to a point and to its complement is enough to recover the difference equation. Secondly, there exists a local Mellin transform analogous to the local Fourier transform. The local Fourier transform describes singularities of a D-module on the affine line in terms of the singularities of its Fourier transform. Similarly, the Mellin transform is an equivalence between D-modules on the punctured affine line and difference modules on the line, and we can relate singularities on both sides via this local Mellin transform. I will also talk about how to apply the same ideas to other kinds of difference equations, such as elliptic equations, which generalize difference and differential equations at once.

• Ronno Das (University of Chicago)

  Points and Lines on Cubic Surfaces

  Abstract: The Cayley-Salmon theorem states that every smooth cubic surface in CP^3 has exactly 27 lines. Their proof is that marking a line on each cubic surface produces a 27-sheeted cover of the moduli space $M$ of smooth cubic surfaces. Similarly, marking a point produces a 'universal family' of cubic surfaces over $M$. One difficulty in understanding these spaces is that they are complements in affine space of incredibly singular hypersurfaces. In this talk I will explain how to compute the rational cohomology of these spaces. I'll then explain how these purely topological theorems have (via the machinery of the Weil Conjectures) purely arithmetic consequences: the typical smooth cubic surface over a finite field F_q contains 1 line and q^2 + q + 1 points.

• Olivier Martin (University of Chicago)

  Voisin's Conjecture on the Gonality of Very General Abelian Varieties

  Abstract: The gonality of a curve C is the minimal degree of a dominant morphism C---> P^1. The (covering) gonality of a variety X is the minimal gonality of the normalization of an irreducible curve in X. In "Chow rings and gonality of general abelian varieties" Voisin studies the Chow group of zero-cycles of very general abelian varieties and deduces a lower bound on the gonality of a very general abelian variety of dimension g. This bound implies that the gonality of a very general abelian variety of dimension g goes to infinity with g, answering affirmatively a question of Bastianelli, De Poi, Ein, Lazarsfeld and Ullery. She also conjectures that a very general abelian variety of dimension greater or equal to 2g-1 has gonality at least g. We will explain a proof of Voisin's conjecture which is based on a generalization of her methods.

• Debaditya Raychaudhury (University of Kansas)

  Very Ampleness and Projective Normality on Higher Dimensional Varieties

  Abstract: In this talk, we will discuss new results on projective normality associated to adjunction line bundles $K_X\otimes L^{\otimes n}$ where $L$ is an ample and globally generated line bundle for varieties with nef canonical bundle in dimension $n\geq 3$. As a corollary, we show some sharp bounds on very ampleness and projectve normality of pluricanonical line bundle in dimenson $3,4,5$ for $K_X$ ample. The case of hyperkähler $n$ folds is of particular interest to us for $4\leq n\leq 10$, here we will sketch the proof of new results on effective very ampleness and projectve normality for an ample and globally generated line bundle $L$.

• Kalila Sawyer (University of Kentucky)

  Calculating Tropical Scrollar Invariants on k-Gonal Chains of Loops

  Abstract: In the quest to understand curves, we often look at their divisors, that is, how many ways we can map them into complex projective space. In particular, we like to study the spaces $W^r_d(C)$ of such maps that have rank r and degree d. The scrollar invariants of a curve give us some notion of how the rank of each divisor changes as we repeatedly add it to itself, which in turn yields some insight into the behavior of $W^r_d(C)$. In this talk we'll introduce and motivate these ideas more carefully and give an overview of how we can use tropical tools to calculate scrollar invariants.

• Fumiaki Suzuki (UIC)

  A remark on a $3$-fold constructed by Colliot-Thélène and Voisin

  Abstract: A classical question asks whether the Abel-Jacobi map is universal among all regular homomorphisms. In this talk, we prove that we can construct a $4$-fold which gives the negative answer in codimension $3$ if the generalized Bloch conjecture holds for a $3$-fold constructed by Colliot-Thélène and Voisin in the context of the study of the defect of the integral Hodge conjecture in degree $4$.

• Dinesh Valluri (University of Western Ontario)

  Essential dimension of parabolic bundles over a non-singular curve

  Abstract: Essential dimension of a geometric object is roughly the number of algebraically independent parameters needed to define the object. In this talk we give upper bounds for the essential dimension of parabolic bundles over a non-singular curve using Borne’s correspondence between parabolic bundles on a curve and vector bundles on a root stack. This is a generalization of the work of Biswas, Dhillon and Hoffmann on essential dimension of vector bundles by using their own techniques.

• Rachel Webb (University of Michigan)

  Virtual Fundamental Classes Via Deformation and Localization

  Abstract: The polynomial $x_1^5 + x_2^5 + . . . + x_5^5$ cuts out a smooth Calabi-Yau hypersurface in $\mathbb{P}^4$, called the quintic threefold. A standard object in Gromov-Witten theory, the moduli space of stable maps to the quintic is a proper Deligne-Mumford stack and carries virtual fundamental class. Recently Chang and Li embedded this moduli of stable maps in the zero section of the moduli space parameterizing maps to $\mathbb{P}^4$ with sections of $\mathcal{O}(5)$. With this view, they gave another construction of the fundamental class that, in many applications, is more useful than the traditional construction. After illustrating Chang and Li's construction, I will present a work joint with Qile Chen and Felix Janda in which we extend Chang-Li's construction to a complete intersection in a smooth projective variety. The argument uses a deformation to the normal cone to reduce the problem to a special case, and then localization to compute that case.

• Wendy Cheng (University of Wisconsin)

  Title: Explicit bound on collective strength of regular sequences of three homogeneous polynomials

  Abstract: In ’Small subalgebras of polynomial rings and Stillman’s conjecture’ by Ananyan and Hochster, it is shown that if f1,··· ,fr ∈ k[x1,··· ,xn] are homogeneous polynomial of degree d, then there exists a bound N = N(r,d) where: if the collective strength of f1,··· ,fr ≥ N, then f1,··· ,fr are regular sequence. In my research, I work on the case where r = 3 and examine how N(3,d) changes with different d. i.e. Does N(3,d) go to infinity as d → ∞? The first interesting case is when r = 3, where we show N(3,2) = 3 and N(3,3) > 3. We are continuing to explore the bound on collective strength of f1,f2,f3 in the case when d goes large.

• Kaelin Cook-Powell (University of Kentucky)

  Title: Irreducible Components of Brill-Noether Loci of General k-Gonal Curves

  Abstract: The classical Brill-Noether Theorem calculates the dimension of Brill-Noether loci of a general curve of genus g. Using tools and techniques from tropical geometry it is possible to calculate the dimension of Brill-Noether loci for general curves of prescribed gonality k, but in general these loci are not equidimensional. However, building upon work of Pfleuger and Jensen-Ranganathan it is possible to guarantee the existence, and calculate the dimension, of certain irreducible components, though this calculation does not exclude the existence of irreducible components of unexpected dimensions.

• Daniel Levine (Penn State University)

  Title: Semistable Chern characters on Del Pezzo surfaces

  Abstract: Abstract: We give an analogue of the Drezet-Le Potier classification of semistable Chern characters on the protective plane for Del Pezzo surfaces with the anti-canonical polarization. Joint work with Shizhuo Zhang.

• Devlin Mallory (University of Michigan)

  Arc closures and the local isomorphism problem

  Abstract: We give an answer in the "geometric" setting to a question of de Fernex, Ein, and Ishii asking when local isomorphisms of k-schemes can be detected on the associated maps of local arc or jet schemes. In particular, we show that their ideal-closure operation (the arc-closure) defined on a local k-algebra (R,m,L) is trivial when R is Noetherian and L is separable over k, and thus that such a germ Spec R has the (embedded) local isomorphism property.

• Michael Perlman (Notre Dame)

  Title: Local cohomology with support in some orbit closures

  Abstract: Let X be a variety with the action of an algebraic group G. Given an orbit, the local cohomology modules with support in its closure encode a great deal of information about its singularities and topology. We will discuss how techniques from representation theory and the theory of D-modules may be used to compute these local cohomology modules in the case when G acts with finitely-many orbits. These techniques will be illustrated via the example of the space of alternating senary 3-tensors with a GL(6)-action. Joint work with András Lőrincz.