Algebraic Geometry Seminar

The complexity of a Calabi-Yau pair (X,B) is an invariant that relates the dimension of X, the Picard rank of X, and the coefficients of B. It was proven by Brown, McKernan, Svaldi and Zong that the complexity of a Calabi-Yau pair is nonnegative, and a variety X admits a Calabi-Yau pair of complexity 0 if and only if X is toric. In this talk we will discuss the geometry of Calabi-Yau pairs of index one and complexity one or two. Both descriptions are done in terms of cluster type varieties, a generalization of toric varieties. In the case of complexity one, we prove that (X,B) is of cluster type. In the case of complexity two, we give a criterion to decide whether the pair is cluster type or not. This is joint work with Joshua Enwright, Jennifer Li and Joaquin Moraga.

In the study of derived categories of coherent sheaves, ample line bundles play a fundamental role -- their tensor powers generate the derived category. This raises a natural question: does this generation property characterize ampleness? The answer is negative, but we show that this categorical property can be checked by a classical numerical criterion naturally extending the Nakai�Moishezon criterion. Moreover, the cone of divisors satisfying this condition lies between the big cone and the ample cone. In this talk I will focus on explaining the case of surfaces, where the geometry becomes especially clear. This is a joint work with Noah Olander.

The ADE classification is an important collection of objects that appears in many fields of mathematics. In the context of algebraic geometry and commutative algebra, the ADE classification proved an important collection of hypersurfaces with unique properties, that were classified by Artin, Du Val, Arnold, Greuel, and many others. In this talk we will discuss the different properties of the ADE classification over fields of both zero and positive characteristic, and how we can generalize these results to over complete regular local rings using a characteristic free approach.

For a projective variety X defined over a finite field, the homology of the space of rational curves on X is closely related to the F_q(T) rational points on X. In particular, Manin's conjecture for the asymptotic count of rational points on a Fano variety suggests a corresponding homological stability conjecture for moduli spaces of rational curves. We will discuss joint work with R. Das, B. Lehmann, and S. Tanimoto establishing both these conjectures in the case where X is a quartic del Pezzo surface.

Rational and Du Bois singularities are among the most important classes of singularities in algebraic geometry because of their nice homological properties and their relation to the singularities of the minimal model program. Recently, there has been a lot of progress in studying their higher analogs. I will talk about recent work (joint with Haoming Ning) on generalizing the notion of higher Du Bois singularities to pairs.

Cubic fourfolds have been classically studied up to birational equivalence, with an eye towards rationality problems. We will discuss two other notions of equivalence: Fourier-Mukai equivalence, and `hyperkahler equivalence'. We'll discuss how these equivalences are conjecturely related. We will give new examples of pairs of cubic fourfolds satisfying all three equivalences. In particular, our examples will be pairs of birational cubic fourfolds, with birational Fano varieties of lines, a previously unknown phenomenon. This is joint work with Corey Brooke and Sarah Frei.

Vanishing theorems are foundational tools in the Log Minimal Model Program, with the Kawamata-Viehweg Vanishing Theorem being one of the most important. However, these fail in positive characteristics. Fano varieties and their log generalizations are expected to behave better in this setting. In this talk, we prove the Kawamata–Viehweg vanishing theorem for surfaces of del Pezzo type over imperfect fields of characteristic p>5 and discuss consequences to threefold singularities.