# Seminars & Events for Algebraic Geometry Seminar

##### Pluricanonical maps on threefolds

We will discuss the problem of effectiveness of Iitaka fibrations for surfaces and threefolds.

##### A Torelli theorem over finite fields

I will discuss a set-theoretic analog of the classical Torelli theorem for curves.

##### Leaves in moduli spaces in characteristic p

We try to understand the geometry of the moduli space of polarized abelian varieties in characteristic p. E.g. the phenomenon that Hecke orbits blow up and down in a rather unpredictable way. Choose a point $x$, corresponding to a polarized abelian variety. We study $C(x)$ consisting of all moduli points of polarized abelian varieties which have the same $p$-adic and $\ell$-adic invariants. This turns out to be a locally closed subset. We discuss properties of these sets, which form a foliation of the related Newton polygon stratum. We give several applications.

##### A Giambelli formula for isotropic Grassmannians

The structure of the cohomology ring of the Grassmannian has been studied for well over a century, but the analogous questions for the non-maximal isotropic Grassmannians of the symplectic and orthogonal groups are rather unexplored. I will describe a new Giambelli type formula which expresses a general Schubert class on a symplectic Grassmannian as a polynomial in certain special Schubert classes, and some applications. This is joint work with Andrew Kresch and Anders Buch, and grew naturally out of our study of the small quantum cohomology ring of these spaces.

##### TBA

##### Vanishing and torsion-free theorems for the log minimal model program

We will discuss Ambro's formulation of Kollár's injectivity, torsion-free, and vanishing theorems. It is indispensable for the study of the log minimal model program for log canonical pairs.

##### Equivalences from geometric $sl_2$ actions

We explain how $sl_2$ actions on derived categories of coherent sheaves can be used to construct new derived equivalences. The example I will describe in detail is an $sl_2$ action via correspondences on the cotangent bundles of Grassmannians which generalizes the basic Mukai flop. More generally we can construct an action on the derived category of coherent sheaves on quiver varieties which lifts Nakajima's action on their cohomology. (joint with Joel Kamnitzer and Anthony Licata)

##### Finiteness theorems for algebraic groups over function fields

If $X$ is a smooth variety over a global field $k$, $G$ is an algebraic group over $k$ equipped with an action on $X$, and $x$ is a point in $X(k)$ then it is natural to ask how the property of $x'$ in $X(k)$ being in the $G(k)$-orbit of $x$ compares with being in the $G(k_v)$-orbit of $x$ for all places $v$ of $k$. In general there is a non-trivial "local-to-global" obstruction space, but one can ask if it is finite. Even when $G$ is semisimple, this finiteness problem leads to the consideration of the isotropy group $G_x$ that is generally not connected or reductive (or even smooth when $char(k)>0$). In the number field case the finiteness of these obstruction spaces was proved by Borel and Serre long ago, but their method used characteristic $0$ in an essential way. Recently in joint work with Gabber and G.

##### Morrison, Mori and Mumford: mirror symmetry, birational geometry, and moduli spaces

Joint Columbia-Courant-Princeton Algebraic Geometry Seminar

##### Convex bodies associated to linear series

Joint Columbia-Courant-Princeton Algebraic Geometry Seminar

##### Towards a classification of modular compactifications of the moduli space of curves

The class of stable curves is deformation-open and satisfies the unique limit property, hence gives rise to the modular Deligne-Mumford compactification of $M_{g,n}$. But the class of stable curves is not unique in this respect; one obtains alternate compactifications by considering, for example, a moduli problem in which elliptic tails are replaced by cusps or in which marked points are allowed to collide. In this talk, we will survey progress toward a systematic classification of these alternate compactifications.

##### Automorphism groups of curves

Hurwitz proved that a complex curve of genus $g>1$ has at most $84(g-1)$ automorphisms. In case equality holds, the automorphism group has a quite special structure. However, in a qualitative sense, all finite groups $G$ behave the same way: the least $g>1$ for which $G$ acts on a genus-$g$ curve is on the order of $(G)\times d(G)$, where $d(G)$ is the minimal number of generators of $G$. I will present joint work with Bob Guralnick on the analogous question in positive characteristic. In this situation, certain special families of groups behave fundamentally differently from others. If we restrict to $G$-actions on curves with ordinary Jacobians, we obtain a precise description of the exceptional groups and curves.

##### Vojta's conjecture on Blowups and GCD Inequalities

Vojta's conjecture is a deep conjecture in Diophantine geometry, implying for example the Bombieri-Lang conjecture and the abc conjecture. In this talk, I will show some cases of the conjecture for blowup varieties. As a consequence, we derive some interesting inequalities of greatest common divisors. An important ingredient will be Schmidt's subspace theorem, both directly and indirectly through the results of Corvaja and Zannier.

##### Algebraic surfaces and hyperbolic geometry

The intersection form on the group of line bundles on a complex algebraic surface always has signature $(1,n)$ for some $n$. So the automorphism group of an algebraic surface always acts on hyperbolic $n$-space. For a class of surfaces including $K3$ surfaces and many rational surfaces, there is a close connection between the properties of the variety and the corresponding group acting on hyperbolic space. (In fancier terms: the Morrison-Kawamata cone conjecture holds for klt Calabi-Yau pairs in dimension 2.)

##### Compactified Jacobians and Abel maps for singular curves

We will discuss the problem of extending the construction of the classical Abel maps for smooth curves to the case of singular curves. The construction of degree-1 Abel maps will be shown, together with an approach for constructing higher degree Abel maps.

##### Arakelov invariants on modular curves

Arakelov theory provides a rich set of invariants. We shall discuss the question of their limiting behavior in several classical examples, with an emphasis on heights of special points and of modular curves.

##### Constructing moduli spaces of objects with infinite automorphisms

Moduli problems parameterizing objects with infinite automorphisms (eg. semi-stable vector bundles) often do not admit coarse moduli schemes but may admit moduli schemes identifying certain non-isomorphic objects. I will introduce techniques to study such moduli stacks and address the question of how such moduli schemes can be intrinsically constructed. The crucial ingredient is the notion of a good moduli space for an Artin stack, which generalizes Mumford's geometric invariant theory and characterizes the desired geometric properties of a moduli scheme parameterizing objects with infinite automorphisms.

##### Vector bundles with sections

Classical Brill-Noether theory studies, for given $g, r, d$, the space of line bundles of degree $d$ with $r+1$ global sections on a curve of genus $g$. We will review the main results in this theory, and the role of degeneration techniques in proving them, and then we will discuss the situation for higher-rank vector bundles, where even the most basic questions remain wide open.

##### Automorphisms mapping a point into a subvariety

Given a variety $X$, a point $x$ in $X$, and a subvariety $Z$ of $X$, is there an automorphism of $X$ mapping $x$ into $Z$? We prove that this problem is undecidable.

##### Calabi-Yau threefolds with vanishing third Betti number

Smooth, projective, three dimensional, algebraic varieties with trivial canonical sheaf and vanishing third etale Betti number do not exist over fields of characteristic zero. In the past few years a number of examples have been found in positive characteristic. Some of these examples and questions they raise will be discussed.