Spring 2019 Abstracts

Jennifer Balakrishnan (Boston University)

Title: Rational points on the cursed curve

Abstract: The split Cartan modular curve of level 13, also known as

the "cursed curve," is a genus 3 curve defined over the rationals. By Faltings' proof of Mordell's conjecture, we know that it has finitely many rational points. However, Faltings' proof does not give an algorithm for finding these points. We discuss how to determine rational points on this curve using "quadratic Chabauty," part of Kim's nonabelian Chabauty program. This is joint work with Netan Dogra, Steffen Mueller, Jan Tuitman, and Jan Vonk.

Gabriele Di Cerbo (Princeton)

Title: Birational boundedness of elliptic Calabi-Yau varieties

Abstract: The minimal model program predicts that, up to a special class of birational equivalences, each projective variety decomposes into iterated fibrations with general fibers of 3 basic types: Fano varieties, Calabi-Yau varieties, and varieties of general type. Our understanding of the boundedness of Fano varieties and varieties of general type is quite solid but Calabi-Yau varieties are still elusive. In this talk, I will discuss recent results on the birational boundedness of elliptic Calabi-Yau varieties with a section. As a consequence, we obtain that there are finitely many possibilities for the Hodge diamond of such varieties. This is joint work with Caucher Birkar and Roberto Svaldi.

Maria Angelica Cueto (Ohio State)

Title: Anticanonical tropical del Pezzo cubic surfaces contain exactly 27 lines.

Abstract: Since the beginning of tropical geometry, a persistent challenge has been to emulate tropical versions of classical results in algebraic geometry. The well-known statement "any smooth surface of degree three in P^3 contains exactly 27 lines'' is known to be false tropically. Work of Vigeland from 2007 provides examples of tropical cubic surfaces with infinitely many lines and gives a classification of tropical lines on general smooth tropical surfaces in TP^3.

In this talk I will explain how to correct this pathology by viewing the surface as a del Pezzo cubic and considering its embedding in P^44 via its anticanonical bundle. The combinatorics of the root system of type E_6 and a tropical notion of convexity will play a central role in the construction. This is joint work with Anand Deopurkar.

Angela Gibney (Rutgers)

Title: Classes on the moduli space of curves from Lie algebras, Gromov Witten theory, and vertex algebras: Identities and generalizations

Abstract: I will introduce classes on the moduli space of curves that arise as Chern classes of Verlinde bundles, constructed from modules over simple Lie algebras, and as the Gromov-Witten loci of smooth homogeneous varieties. These classes are equivalent in the most basic cases. Examples, conjectures, and generalizations using conformal vertex algebras will be presented. This talk will be about joint work with Prakash Belkale, Chiara Damiolini, and Nicola Tarasca.

Edray Goins (Pomona College)

Title: Monodromy groups of compositions of Belyi maps

\beta: \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C)
n
S_n
\text{Mon}(\beta)

Abstract: Given a Belyi map of degree , it is well known that its monodromy group is a subgroup of the symmetric group . In fact, this group can be viewed as the ``Galois closure'' of the automorphism group

\text{Aut}(\beta) \subseteq S_n
\beta \circ \phi:\mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C)
\beta

. Given a composition of two Belyi maps and

\gamma
\text{Mon}(\beta \circ \gamma) \subseteq \text{Mon}(\gamma) \sim \text{Mon}(\beta)

, it is known that the monodromy group is contained in the wreath product of the monodromy groups of each of the maps. However, when do we have equality? And what exactly is the relationship between these three groups?

In the 2018 doctoral thesis of Jacob Bond, there was a description of the relationship between these three. Explicitly,

\text{Mon}(\beta \circ \gamma) \simeq \rho_\gamma(A) \sim \text{Mon}(\beta)
\rho_\gamma(A)
E_\beta \to \pi_1( X, x) \to \text{Mon}(\gamma)

where is a subgroup of the collection of maps from the edges of the Dessin d'Enfant of

\beta
X = \mathbb P^1(\mathbb C) \backslash \{ 0, \, 1, \, \infty \}
\gamma

to the Fundamental Group of the thrice punctured sphere to the monodromy group of . In this talk, we explain the details, and provide some examples. This is joint work with Jacob Bond.

Eyal Markman (University of Massachusetts)

Title: The Hodge conjecture for the generic abelian fourfold of Weil type with discriminant 1.

Abstract: A generalized Kummer variety of dimension 2n is the fiber of the Albanese map from the Hilbert scheme of n+1 points on an abelian surface to the surface. We compute the monodromy group of a generalized Kummer variety via equivalences of derived categories of abelian surfaces. As an application we prove the Hodge conjecture for the generic abelian fourfold of Weil type with complex multiplication by an arbitrary imaginary quadratic number field K, but with trivial discriminant invariant. The latter result is inspired by a recent observation of O'Grady that the third intermediate Jacobians of smooth projective varieties of generalized Kummer deformation type form complete families of abelian fourfolds of Weil type.

Angelo Vistoli (Scuola Normale Superiore)

Title: Essential dimension of algebraic groups

Abstract: Essential dimension of an algebraic group G is a fundamental invariant of G that measures the complexity of G-torsors. I will survey a few of the known results, particularly for finite groups, stressing the depth of our ignorance.