# Seminars & Events for Algebraic Geometry Seminar

##### The geometric constants in Manin's Conjecture

Manin's Conjecture predicts that the growth of points of bounded height is controlled by certain geometric constants. I will analyze the geometry underlying these constants and discuss applications to Manin's Conjecture. A key tool is the minimal model program.

##### Local volumes and equisingularity theory

The epsilon multiplicity of a module is a natural generalization of the Hilbert-Samuel multiplicity of an ideal. It can be expressed as the volume of a suitable Cartier divisor associated with the module. In this talk, I will present a result that determines the change of the epsilon multiplicity of a module across flat families. The change is the multiplicity of the polar curve associated with the module. This result generalizes previous work of Gaffney, Kleiman, Teissier and Hironaka. I will discuss various applications of this relation, among which to the Whitney-Thom equisingularity theory and the topology of isolated singularities.

##### On Minkowski bases for Newton-Okounkov bodies

The Newton-Okounkov bodies of linear series on an n-dimensional projective variety is a compact convex body in real n-space which carries information about the linear series. However, in general it is hard to determine in practice. We show that under certain conditions there exist simple "building blocks" for al Newton-Okounkov bodies of a given variety, a so called Minkowski basis. Additionally, we establish a consequence of the existence of a Mikowski basis for the shape of the global Okounkov body studied by Lazarsfeld and Mustata.

##### Toric degenerations and symplectic geometry of projective varieties

I will explain some recent general results about symplectic geometry of projective varieties using toric degenerations (motivated by commutative algebra and the theory of Newton-Okounkov bodies). The main result is the following: Let X be a smooth n-dimensional complex projective variety equipped with an integral Kahler form. We show that for any epsilon>0, the manifold X has an open subset U (in the usual topology) such that vol(X)-vol(U)< epsilon, and moreover U is symplectomorphic to the algebraic torus (C*)^n equipped with a "toric" Kahler form. The proof is based on the construction of a toric degeneration of X. As applications we obtain lower bounds on the Gromov width of X. We also get a full symplectic ball packing of X by d balls of capacity 1 where d is the degree of X.

##### Effective Matsusaka's theorem for surfaces in characteristic p

The goal of this talk is to explain how to obtain an effective version of Matsusaka's theorem for arbitrary smooth algebraic surfaces in positive characteristic. It provides an effective bound on the multiple which makes an ample line bundle very ample. The proof is based on a Reider-type analysis of adjoint linear series combined with bend and break techniques

##### Syzygies on abelian surfaces, construction of singular divisors, and Newton-Okounkov bodies

Constructing divisors with prescribed singularities is one of the most powerful techniques in modern projective geometry, leading to proofs of major results in the minimal model program and the strongest general positivity theorems by Angehrn-Siu and Kollár-Matsusaka. We present a novel method for constructing singular divisors on surfaces based on infinitesimal Newton-Okounkov bodies. As an application of our machinery we discuss a Reider-type theorem for higher syzygies on abelian surfaces building on earlier work of Lazarsfeld-Pareschi-Popa.

##### Deformation of quotient singularities

##### Non-Archimedean geometry in rank >1

Recent work of Nisse-Sottile, Hrushovski-Loeser, Ducros, and Giansiracusa-Giansiracusa has demonstrated that valuation rings of rank >1 play an important role in the geometry of analytic and tropical varieties over non-Archimedean valued fields of rank 1. In this talk, I will present recent work with Dhruv Ranganathan in which we prove several foundational results on the geometry of analytic and tropical varieties over higher rank valued fields, and recent work with Max Hully in which we use rank 2 valuations to give a new, non-analytic proof of Rabinoff's theorem on the correspondence between tropical and algebraic intersection multiplicities.

##### Mori fibre spaces for 3-folds in positive characteristic

There has been much progress in recent years on the LMMP for 3-folds in characteristic p>5. In this talk I will discuss the proof of the base point free theorem and how it leads to termination of the LMMP with scaling and the existence of Mori fibre spaces. This is joint work with Caucher Birkar.

##### Hilbert scheme of points on singular surfaces

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.

##### Interpolation for normal bundles of general curves

This talk will address the following question: When does there exist a curve of given degree d and genus g, passing through n general points p_1, p_2,..., p_n in P^r?

##### Ample divisors on Hilbert schemes of points on surfaces

The Hilbert scheme of n points on a smooth projective surface X is a smooth projective variety due to a classical result of Fogarty. A natural question about these spaces is to determine their ample divisors. Using techniques from derived categories developed by Bayer and Macrì, we describe the nef cones if X has Picard rank 1, irregularity 0 and n is large. Moreover, we compute the nef cone if X is the blow-up of the projective plane at 8 general points for any n. This is joint work with Bolognese, Huizenga, Lin, Riedl, Woolf, and Zhao originating from the boot camp of the 2015 Algebraic Geometry Summer Institute in Utah.

##### Automorphisms of Blowups

We use p-adic analytic methods to analyze automorphisms of smooth projective varieties. We prove a version of the dynamical Mordel-Lang conjecture for arbitrary subschemes of a variety. We apply this result to (1) classify automorphisms of X for which there exists a divisor D whose intersection with its iterates are not dense in D, and (2) show that various properties of Aut(X) (for example finiteness of its component group) are not altered by blowups in high codimension. This is joint work with John Lesieutre.

##### Intersection multiplicity over a two-dimensional base

Let X be smooth over a regular, two-dimensional base scheme Y. We show that properly-meeting cycles on X intersect with positive multiplicity (in the sense of Serre's Tor-formula). When Y is one-dimensional - say, the ring of integers of a number field - we use these techniques to investigate the extent to which intersection multiplicities can detect transversality.

##### Adjoint dimension of foliations

**PLEASE NOTE SPECIAL TIME.** The classification of foliated surfaces by Brunella, McQuillan and Mendes carries many similarities with Enriques-Kodaira classification of surfaces but also many important differences. I will discuss an alternative classification scheme where the role of differential forms along the leaves is replaced by differential forms along the leaves with values in fractional powers of the conormal bundle of the foliation. In this alternative setup one obtains a classification of foliated surfaces closer to the usual Enriques-Kodaira classification. If time permits, I will show how to apply this alternative classification to describe the Zariski closure of the set foliations which admit rational first integral of bounded genus in families of foliated surfaces. Joint work with Jorge Vitorio Pereira.

##### TBA - Kevin Tucker

##### Ulrich bundles and variants on ACM surfaces

**Please note different day. ** A sheaf F on a polarized variety (X,O_X(1)) is ACM if F(k) has no intermediate cohomology for any integer k. The variety X is ACM if O_X is ACM. One important class of ACM bundles is the class of Ulrich bundles. An Ulrich bundle is a globally generated ACM bundle which has the largest possible space of global sections in a certain sense. Spinor bundles on quadrics are a familiar example of Ulrich bundles. It is an important problem to understand whether or not a smooth projective variety admits an Ulrich bundle, even in dimension two. In this talk, I will describe striking links between Ulrich bundles and problems in commutative algebra and representation theory. I will also discuss some partial progress on the existence problem with R. Kulkarni and Y.

##### Measures of irrationality for hypersurfaces of large degree

The gonality of a smooth projective curve is the smalles 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 notions, and present joint work with L. Ein and R. Lazarsfeld. Our main rezult is thet 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.

##### Frobenius semisimplicity for convolution morphisms

This article concerns properties of mixed ℓ-adic complexes on varieties over finite fields, related to the action of the Frobenius automorphism. We establish a fiberwise criterion for the semisimplicity and Frobenius semisimplicity of the direct image complex under a proper morphism of varieties over a finite field. We conjecture that the direct image of the intersection complex on the domain is always semisimple and Frobenius semisimple, this conjecture would imply that a strong form of the decomposition theorem of Beilinson-Bernstein-Deligne-Gabber is valid over finite fields.

##### Cohomology jump loci and examples of NonKahler manifolds

Cohomology jump loci are homotopy invariants associated to topological spaces of finite homotopy type. They are generalizations of usual cohomology groups. I will give a survey on the theory of cohomology jump loci of projective, quasi-projective and compact Kahler manifolds, due to Carlos Simpson, Nero Budur and myself. In the second part of the talk, I will introduce some concrete examples of 6-dimensional symplectic-complex Calabi-Yau manifold, which satisfies all the known topological criterions of compact Kahler manifolds such as Hodge theory and Hard Lefschetz theorem, but fail the cohomology jump locus property of compact Kahler manifolds. The second part is joint work with Lizhen Qin.