Algebraic Geometry Seminar

We discuss two vanishing theorems for mixed Hodge modules, one of Esnault-Viehweg type and the other of Kawamata-Viehweg type.

Compact hyperkahler varieties are higher-dimensional analogues of K3 surfaces. Very few examples are known: two infinite series and two sporadic examples. At the same time we cannot prove that there are not many more examples. In 2013, Lehn et al. constructed a family of hyperkahler 8-folds Z from the variety of twisted cubic curves on a cubic 4-fold. I will discuss some joint work with Lehn from 2014 and some new work, which together clarify Z's place in the moduli space of hyperkahlers. In particular it is deformation-equivalent, but typically not isomorphic or even birational, to the Hilbert scheme of 4 points on a K3 surface. A crucial role is played by Kuznetsov's K3 subcategory of the derived category of the cubic.

We show that a large class of complex hypersurfaces in all dimensions are not stably rational. (A variety is stably rational if its product with some projective space is birational to projective space.) Namely, for all d at least about 2n/3, a very general hypersurface of degree d in P^{n+1} is not stably rational. The result covers all the degrees in which Kollar proved that a very general hypersurface is not rational, and a bit more. For example, very general quartic 4-folds are not stably rational, whereas it was not even known whether these varieties are rational.

The Hilbert scheme of points on a quasi-projective variety parameterizes its zero-dimensional subschemes. These Hilbert schemes are smooth and irreducible for smooth surfaces but will eventually become reducible for sufficiently singular surfaces. In this talk, I provide the first class of examples of singular surfaces whose Hilbert schemes of points are irreducible, namely surfaces with at worst cyclic quotient rational double points. I will also describe some consequent geometric properties of these irreducible Hilbert schemes.

This is a classical fact that if a plane Cremona map F:P^2-->P^2 is given by Polynomials of degree d then so does its inverse. A generalization of this fact for birational maps from P^n to P^n is due to Gabber who showed in 80's that the inverse map has degree at most d^{n-1}. However if one considers an arbitrary projective variety X of P^n, then bounding the degree of the inverse map becomes a subtle challenge. This is the leitmotiv of this talk which answers a question by Jeremy Blanc. We start from Plane Cremona maps to show how the Betti table of the base ideal constrains birationality. Moving from Plane case to higher dimensions, we encounter the role of higher polynomial equations of the base ideal, consequently the structure of the Rees algebra appears. A new Criterion for Birationality is presented which provides information on the inverse map for any birational map F:X-->Y of projective varieties.

The dynamical Mordell--Lang conjecture predicts that if f is an endomorphism of a complex variety X, with p a point of X and V a subvariety, then the set of n for which f^n(p) lands in V is a union of a finite set and finitely many arithmetic progressions. When f is \'etale, this is a result of Bell--Ghioca--Tucker. I'll discuss an extension of this result to the setting in which p and V are non-reduced closed subschemes of X, and show how this statement can be applied to study dynamically interesting (e.g. positive entropy) automorphisms of complex varieties in higher dimensions. This is joint work with Daniel Litt.

In 1922, Mordell showed that the set of rational points on an elliptic curve over Q has the structure of a finitely generated abelian group. Surprisingly, there are only 15 possibilities for the torsion part of this group; this is a spectacular result of Mazur from 1977. In this talk I will discuss some higher-dimensional generalizations of results along Mazur's theorem, including recent joint work with Dan Abramovich on abelian varieties, and on-going work for K3 surfaces, where I will argue that a suitable analogue of the group of rational points is the Brauer group.

The gonality of a smooth projective curve is the smallest degree of a map from the curve to the projective line. There are a few different definitions that attempt to generalize the notion of gonality to higher dimensional varieties. The intuition is that the higher these numbers, the further the variety is from being rational. I will discuss some of these definitions, and present joint work with Lawrence Ein and Rob Lazarsfeld. Our main result is that if X is an n-dimensional hypersurface of degree d at least 5/2 n, then any dominant rational map from X to P^n must have degree at least d-1.

Faltings' theorem states that curves of genus g>1 have finitely many rational points. Using the ideas of Faltings, Mumford, Parshin and Raynaud, one obtains an upper bound on the number of rational points, but this bound is too large to be used in any reasonable sense. In 1985, Coleman showed that Chabauty's method, which works when the Mordell-Weil rank of the Jacobian of the curve is smaller than g, can be used to give a good effective bound on the number of rational points of curves of genus g > 1. We draw ideas from nonarchimedean geometry to show that we can also give an effective bound on the number of rational points outside of the special set of the d-th symmetric power of X, where X is a curve of genus g > d, when the Mordell-Weil rank of the Jacobian of the curve is at most g-d.

Quantum sheaf cohomology arises as the OPE ring of A/2 twisted theories with (0,2) supersymmetry. It is a deformation of the ordinary quantum cohomology. Its structure is well understood for toric varieties, but few results exsited for other spaces. In this talk, I will introduce recent progress on the study of quantum sheaf cohomology on Grassmannians with a deformed tangent bundle, especially an algebraic treatment of the classical limit, known as polymology, and how quantum corrections can be inferred.

Many “pathologies�of varieties in characteristic p do not occur if the variety in question lifts to characteristic zero, or at least if it lifts modulo p^2. We will review classical and recent results on liftable varieties, discuss liftability of Frobenius split varieties, and present some new (quite elementary) examples of non-liftable varieties. This is joint work with Maciej Zdanowicz.

The Enriques-Kodaira classification of surfaces is one of the most celebrated accomplishments toward understanding algebraic varieties. A detailed classification of surfaces of general type, however, seems to be very difficult. Several cases of this kind are known in characteristic 0. In my talk I would like to present a classification result for surfaces of general type in positive characteristics, and in particular we work on surfaces with Euler characteristic 1 and Albanese dimension 4. This project is inspired by a paper of Hacon and Pardini but contains a lot of new ideas, including the construction of a generic vanishing theorem for surfaces that lift to W_2(k), the second Witt vectors.

The algebraic cobordism theory constructed Morel and Levine is a universal oriented Borel-Moore homology theory for schemes. Levine and Pandaripande developed an equivalent algebraic cobordism theory and many results using degeneration methods in algebraic geometry can be understood as invariants of this theory. In this talk I will talk about the generalization of the algebraic cobordism theory to bundles and divisors on varieties and discuss its application to count singular curves with tangency conditions in varieties of any dimension. This enumeration is motivated by Caporaso and Harris' recursive formula for nodal curves with fixed tangency multiplicities with a line on the complex projective plane.