Amherst 2016‎ > ‎


Nicolas Addington. Some new rational cubic 4-folds.

Cubic 4-folds lie at the intersection of two lines of inquiry: on the one hand they're the simplest hypersurfaces whose rationality is not understood, having resisted both old and new attempts to show that irrational ones exist; on the other hand, they have a beautiful connection to K3 surfaces and hyperkähler geometry.  I will exhibit a new family of cubic 4-folds: they are fibered in sextic del Pezzo surfaces, they have associated K3 surfaces of degree 2 with 3-torsion Brauer classes, and when the Brauer class vanishes, they yield new examples of rational cubics for the first time in two decades.  Joint with Hassett, Tschinkel, and Várilly-Alvarado.

Asher Auel. Stable rationality of conic bundle fourfolds.

The conic bundle threefolds of Artin and Mumford were the first examples of rationally connected threefolds that were not stably rational.  In the last few years, a breakthrough in the stable rationality problem was made possible by a degeneration method for the universal triviality of the Chow group of 0-cycles.  This method, initiated by Voisin and developed by Colliot-Thélène and Pirutka, requires the construction of singular varieties having nontrivial unramified cohomology as well as well-behaved resolutions of singularities.  I will explain a new geometric construction of conic bundle fourfolds satisfying these properties, yielding new families of rationally connected fourfolds whose very general member is not stably rational.  The construction uses the theory of contact surfaces and the geometry of special arrangements of rational plane curves, as well as matrix factorizations and symmetric arithmetic Cohen-Macaulay sheaves.  This is joint work with Christian Böhning, Hans-Christian Graf von Bothmer, and Alena Pirutka.

Ana-Maria Castravet. The derived category of moduli spaces of pointed stable rational curves.

I will report on joint work with Jenia Tevelev on Orlov's question on exceptional collections on moduli of pointed stable rational curves and related spaces.

Daniel Huybrechts. The global Torelli theorem for cubic fourfolds via Kuznetsov's K3 category.

This is joint work with Jorgen Rennemo. After recalling what is known about the Global Torelli theorem for smooth hypersurfaces, I'll explain an approach towards the Global Torelli theorem for smooth cubic hypersurfaces of dimension four (a theorem originally due to Claire Voisin) that relies on Kuznetsov's category A_X and the derived Global Torelli theorem for K3 surfaces. Although motivated by results on the category of graded matrix factorizations, the talk will mostly stay in the realm of Fourier-Mukai functors. No prior knowledge of A_X is necessary.

Aaron Pixton. Graph-counting relations in the tautological ring.

The tautological ring R^*(M_g) is the subring of the Chow ring of the moduli space of smooth curves of genus g generated by the kappa classes. The lowest degree relations in the tautological ring, in R^d(M_{3d-1}), can be interpreted as counting cubic graphs on 2d vertices. I will explain how to generalize this graph-counting perspective to generate many more tautological relations (conjecturally all of them).

Colleen Robles. Characterization of Gross's Calabi-Yau type variations of Hodge structure by characteristic forms.

Gross showed that to every Hermitian symmetric tube domain we may associate a canonical variation of Hodge structure (VHS) of Calabi-Yau type. The construction is representation theoretic, not geometric, in nature, and it is an open question to realize this abstract VHS as the variation induced by a family of polarized, algebraic Calabi-Yau manifolds.

In order for a geometric VHS to realize Gross's VHS it is necessary that the invariants associated to the two VHS coincide.  For example, the Hodge numbers must agree.  The latter are discrete/integer invariants. Characteristic forms are differential-geometric invariants associated to VHS (introduced by Sheng and Zuo).

Remarkably, agreement of the characteristic forms is both necessary and sufficient for a geometric VHS to realize one of Gross's VHS.  That is, the characteristic forms characterize Gross's Calabi-Yau VHS.  I will explain this result, and discuss how characteristic forms have been used to study candidate geometric realizations of Gross's VHS.

David Treumann. Betti spectral curves.

I will discuss work with Shende, H. Williams, and Zaslow on the moduli space of Lagrangian surfaces in a symplectic 4-manifold.  When the 4-manifold is a cotangent bundle, these spaces generalize the moduli of locally constant sheaves on the base, and one might think of the Lagrangian surfaces as being Betti analogs of spectral curves.  Exact Lagrangians play a special role, so that analyzing our moduli space gives a lower bound, often infinite, for the number of them.

Giancarlo Urzúa. On geography and moduli of surfaces of general type.

I will start by surveying recent results on Chern slope geography (joint with X. Roulleau in char=0 and with R. Codorniu in char>0), then pass to some results (joint with J. Rana and J. Tevelev, and S. Coughlan) on p_g=0 surfaces, and open questions about their moduli spaces. The Lee-Park connection between moduli spaces and geography will bring into the picture the explicit minimal model program for degenerations of surfaces (joint with P. Hacking and J. Tevelev), and the possible quotient singularities of surfaces at the boundary of the compactified moduli space (joint with A. Stern). In particular I will discuss sharp bounds on the index of these singularities for fixed K^2 (joint with J. Rana), explaining results, some optimal examples, and open problems.