Algebraic Geometry Seminar

The theory of stable log maps was developed recently by Abramovich-Chen-Gross-Siebert to generalize the theory of relative stable maps. To illustrate the idea of using log geometry, I will focus on the basic situation when the target is given by a variety with a Cartier divisor. Stable log maps relative to more complicated boundary, for example simple normal crossings or even toric, can be constructed in a similar manner. This is based on a joint work with Dan Abramovich.

Despite intensive investigation over the years and its innocuously classical appearance, there remain some tantalizing open questions regarding the birational geometry of the moduli space of marked rational curves. One such question, dating from a 2000 paper of Hu and Keel, is to determine whether \(\bar{M}_{0,n}\) is a Mori dream space. Roughly speaking, this would say that its Mori-theoretic information is completely determined by variational GIT in a natural way. In this talk I will discuss joint work with Dave Jensen and Han-Bom Moon in which we construct a wide range of birational models using a GIT construction inspired by two constructions of Kapranov from 1993. These models include \(\bar{M}_{0,n}\) itself as well as all the Hassett weighted models. A consequence is that we exhibit explicit flips between models and give a glimpse of the Mori dream behavior envision by Hu and Keel.

A basic question in arithmetic geometry is whether a given variety defined over a non-closed field admits a rational point. When the base field is of geometric nature, i.e., function fields of varieties, one naturally hopes to solve the problem via purely geometric methods. In this talk, I will discuss the geometry of the moduli space of sections of a projective homogeneous space fibration over an algebraic curve and its MRC quotient. By the results of Esnault and Graber-Harris-Starr, it leads to answers for the existence of rational points on projective homogeneous spaces defined either over a global function field or over a function field of an algebraic surface.

The Cox Ring of an algebraic variety is a generalization of the homogeneous coordinate ring of a projective variety. I will give an introduction to Cox Rings and how they are used in birational geometry, as well as some ideas from the minimal model program, with the aim of showing that the Cox ring of a Fano variety over the complex numbers is Gorenstein.

The famous Yau-Tian-Donaldson conjecture says that a polarized variety (X,L) admits a constant scalar curvature metric if and only if it is K-polystable. The last notion is a completely algebraic notion which I will concentrate on in this talk. More precisely, we will study the K-stability question of Fano varieties. Our new point is that we will put in the machinery of the Minimal Model Program to modify the test configuration and show that the DF-invariants are decreasing. In particular, we answer a conjecture of Tian under the assumption that the Picard number is 1. This is a joint work with Chi Li.

Mildly singular Fano varieties of Picard number one are important objects to study from the minimal model program point of view. For the set of three dimensional \(\epsilon\)-klt log \(\mathbb Q\)-Fano pairs \((X,\Delta)\) of Picard number one, we show that there is a volume bound \(-(K_X+\Delta)^3\leq M(3,\epsilon)\) depending only \(\epsilon\). This result is related to the Borisov-Alexeev-Borisov conjecture which asserts boundedness of the set of n-dimensional \(\epsilon\)-klt log \(\mathbb Q\)-Fano varieties.

Counting curves in projective spaces that pass through various linear subspaces and that are tangent to various hyperplanes (or hypersurfaces) is a classical theme in algebraic geometry. An example is there are 3264 conics tangent to 5 general conics in the projective plane. Several fundamental problems in this area remained unsolved until the advent of Kontsevich moduli space of stable maps. In this talk, I will discuss how to use this tool to count genus one space curves.

In this talk, we are going to prove a uniformity result for the Iitaka fibration f from X to Y, provided that the generic fiber has a good minimal model and the variation of f is zero or that the Kodaira dimension of X is equal to the dimension of X minus 1.

For a nonsingular projective variety X of general type, it's known that |mK_X| induces a birational map for any m sufficiently large. It's an important problem to bound this integer m. In this talk, we will show that, in positive characteristic, |4K_X| is birational providing that X has maximal Albanese dimension.

In this talk, I will present a derived version of the resolutions of the moduli spaces of stable maps. This resolution can be used to rigorously define the so-called reduced GW numbers of CY threefolds (i.e., the GW numbers associated to the main components of the stable map moduli).

The derived resolutions are singular (in the usual sense). For further applications, a resolution (in the usual sense) is desirable; I will describe how to achieve this by a sequence of modular blowups in the case of genera 1 and 2.

Moduli spaces of semi-stable real and quaternionic vector bundles of fixed topological type over a smooth real algebraic curve can be expressed as Lagrangian quotients and embedded into the symplectic quotient corresponding to the moduli variety of semi-stable algebraic vector bundles of fixed rank and degree on the complexified curve. When the rank and degree are coprime, these Lagrangian quotients are connected components of the real locus of the complex moduli variety endowed with the real structure induced from the real structure of the complex curve. The presentation as a quotient enables us to generalize the methods of Atiyah and Bott to a setting with involutions, and compute the mod 2 Poincare polynomials of these moduli spaces of real and quaternionic vector bundles in the coprime case. This is based on joint work with Florent Schaffhauser.

Donaldson-Thomas theory provides a virtual count of curves on a smooth Calabi-Yau threefold X. When X is singular, Donaldson-Thomas theory is not defined. However, when X is the coarse moduli space of an orbifold, there are two candidates for producing virtual counts related to X: virtual counts on the orbifold itself, and virtual counts on a crepant resolution of X. The Donaldson-Thomas crepant resolution conjecture states that these two approaches are equivalent. In this talk, I will present progress in proving this conjecture by introducing the intermediate counting theory of tilted pairs.

A central problem in curve theory is to describe algebraic curves in a given projective space with fixed genus and degree. One wants to know the extrinsic geometry of the curve, i.e information on the equations defining the curve. Koszul cohomology groups in some sense carry 'everything one wants to know' about the extrinsic geometry of curves in projective space: the number of equations of each degree needed to define the curve, the relations between the equations, etc. In this talk, I will present a new method using deformation theory to study Koszul cohomology of general curves. Using this method, I will describe a way to determine number of defining equations of a general curve in some special degree range (but for any genus).

Consider the locus L of curves inside the moduli space of smooth curves admitting a map to the projective line with prescribed ramification profile over two points. This geometric condition can be expressed in two equivalent ways, either as a Hurwitz space, or by intersecting sections of the universal Jacobian. Each gives rise to a Chow class that corresponds to some closure of L inside some partial compactification of the moduli space of curves. In this talk, I will discuss how these classes compare, how they may be expressed in the tautological ring (thanks to recent work of Hain, and Grushevsky-Zakharov), and how this comparison may possibly relate to other results in Hurwitz theory. This is joint work with Renzo Cavalieri and Jonathan Wise and will be, for the most-part, an easy-going continuation of my talk from last year.

Let E be a rank two vector bundle over a Riemann surface C. The moduli space of stable pairs on E, in the sense of Pandharipande-Thomas, provides an alternative to the space of stable maps to E for the purpose of "counting" curves in the threefold E. The scaling action on E induces a torus action on the stable pairs moduli space. The moduli spaces of torus fixed stable pairs can be described as closed subschemes of products of quotient schemes of symmetric powers of E. The description is somewhat compatible with the obstruction theories. In favourable situations this can be used to express stable pairs invariants in terms of quotient scheme invariants---the latter being well-understood. In the "favourable situations" one thus obtains "explicit formulas" for the full descendant stable pairs theory of E.

Nash formulated this problem in an attempt to understand resolution of singularities of a variety X in relation with the space of arcs in X centered at the singular locus. The space of arcs is an infinite dimensional algebraic variety given by the inverse limit of the spaces of n-jets, which are finite dimensional algebraic varieties. Consider a resolution of singularities of X and take the decomposition of the exceptional divisor \(E = \cup_i E_i\). Given any arc \(\gamma \colon (\mathbb{C}, 0) → (X, SingX)\) one can consider the lifting \(\gamma \colon (\mathbb{C}, 0) → (X, E)\). Nash considered the set of arcs whose lifting \(\gamma\) meets a fix divisor \(E_i\) , that is \(\gamma(0) \in E_i\) and proved that these are irreducible sets of the space of arcs. Nash's question is whether for the essential divisors \(E_i\) they are in fact irreducible components of the space of arcs or not (an essential divisor appears by definition in any resolution of X up to birational mapping). He conjectured that the answer was yes for the case of surfaces (for which there exists a minimal resolution that has only essential divisors) and suggested the study in higher dimensions. In 2003, Ishii and Kollár gave an example of a variety of dimension 4 for which some of these sets are not. Hence the case of dimension 2 and 3 remained opened.

Recently we solved the conjecture for the surface case in a joint work with J. Fernández de Bobadilla. I will give an introduction to the problem and details a of the proof for the normal surface case. After works of M. Lejenune Jalabert, A. Reguera and J. Fernández de Bobadilla the problem deals with holomorphic 1-parameter families of convergent arcs. The key of our approach is to work with representatives of appropriate arc families and find a topological obstruction to their existence. The obstruction is expressed as a bound for the euler characteristique of the normalization of the representative of the generic member of the family, which we know is a disc.

Ulrich bundles occur naturally in a variety of algebraic and algebro-geometric topics, including determinantal and Pfaffian descriptions of hypersurfaces, the computation of resultants, and the representation theory of generalized Clifford algebras. In this talk I will discuss the connection between the existence of rank r Ulrich bundles on a degree-d del Pezzo surface X, the geometry of curves of degree dr on X, and points on these curves---and how del Pezzo surfaces are the only arithmetically Gorenstein surfaces for which this connection can hold. This is joint work with Emre Coskun and Rajesh Kulkarni.

The Hodge theorem is one of the most important results in complex geometry. It asserts that for a complex projective variety X the topological invariants \(H^*(X, \mathbb{C})\) can be refined to new ones that reflect the complex structure. The traditional statement and proof of the Hodge theorem are analytic. Given the multiple applications of the Hodge theorem in algebraic geometry, for many years it has been a major challenge to eliminate this analytic aspect and to obtain a purely algebraic proof of the Hodge theorem. An algebraic formulation of the Hodge theorem has been known since Grothendieck's work in the early 1970's. However, the first purely algebraic (and very surprising) proof was obtained only in 1991 by Deligne and Illusie, using methods involving reduction to characteristic p. In my talk I shall try to explain their ideas, and how recent developments in the field of derived algebraic geometry make their proof more geometric.

A classical theorem of Kodaira states that ample line bundles are characterized by the positivity of their curvature form. More generally, one expects that the geometric "positivity" of a line bundle L can be detected on the metrics carried by L. The key tool relating these two concepts is the multiplier ideal. I will introduce multiplier ideals and explain how to obtain bounds on the behavior of analytic multiplier ideals using algebraic constructions.

Given a projective variety X of dimension d, a "flag" of subvarieties Y_i, and a big divisor D, Okounkov showed how to construct a convex body in R^d, and in the last few years, this construction has been developed further in work of Kaveh-Khovanskii and Lazarsfeld-Mustata. In general, the Okounkov body is quite hard to understand, but when X is a toric variety, it is just the polytope associated to D via the standard yoga of toric geometry. I'll describe a more general situation where the Okounkov body is still a polytope, and show that in this case X admits a flat degeneration to the corresponding toric variety. As an application, I'll describe some toric degenerations of flag varieties and Schubert varieties, and explain how the Okounkov bodies arising generalize the Gelfand-Tsetlin polytopes.

Hilbert schemes provide a useful tool for moduli problems, but are difficult to explicitly describe in most situations. In my talk, I will discuss a specific example, namely the Hilbert scheme of degree 12 Fano threefolds. Among its many irreducible components, there are four special components which correspond to different families of smooth Fano threefolds. I will describe the geometry of these special components and their intersection behavior. Motivation coming from mirror symmetry for studying this particular Hilbert scheme will also be discussed. This project is joint work with J. Christophersen.

Castelnuovo-Mumford regulairy of varieties has drawn considerable attention in recent twenty years. There are two main general results about smooth curves and surfaces. Gruson-Peskine-Lazarsfeld showed that for a smooth curve X, one has \(reg X \leq deg X - codim X+1\). And Lazarsfeld showed that for a smooth surface the above result is still true. This regularity bound was formulated and conjectured by Eisenbud-Goto for any variety. In this talk we use duality theory to first give a quick proof for smooth curves and surfaces and then prove this bound for normal surfaces which have rational, Gorenstein elliptic or log canonical singularities. This is joint work with Lawrence Ein.

To probe the infinitesimal structure of a moduli space of geometric objects, one seeks to understand families of those objects over "fat points". Remarkably, these deformation problems tend to admit cohomological solutions of a common form: obstructions in H^2, deformations in H^1, and automorphisms in H^0. I will offer an explanation for this common form, coming from some exotic Grothendieck topologies. We will see how this point of view works in several examples. No prior knowledge about Grothendieck topologies or deformation theory will be assumed.