# Seminars & Events for Algebraic Geometry Seminar

##### TBA - Lieblich

##### On the moduli space of quintic surfaces

We describe the use of GIT and stable replacement for studying the geometry of a special compactification of the moduli space of smooth quintic surfaces, the KSBA compactification. In particular we discuss the interplay between non-log-canonical singularities and boundary divisors. Our presentation will be enriched by drawing analogies with similar phenomena on the moduli space of curves of genus three.

##### Flag Hilbert Schemes

Just like the punctual Hilbert scheme of a smooth surface parametrizes ideals supported at a given point, the flag Hilbert scheme parametrizes such ideals together with a full flag to the structure sheaf. It can be thought of as an lci counterpart of the variety of commuting upper triangular matrices, and as such is relevant for partially resolving the singularities of the punctual Hilbert scheme. We recently found applications of this variety to torus knot invariants and the representation theory of the Hilbert scheme, and a number of people are looking into using it to better understand categorical braid invariants. Joint work with Eugene Gorsky and Andrei Okounkov.

##### Strictly pseudo-effective classes

The pseudo-effective cone is the closure of the convex cone of effective cycle classes of arbitrary fixed (co)dimension in the numerical group of a projective variety. Of particular interest are the boundary classes that were added upon closing. A recent conjecture by Debarre, Jiang, and Voisin aims to bring geometry to such classes that pushforward to zero by a morphism to another projective variety. I will present progress on this and related results coming from joint work with Brian Lehmann.

##### Deformations of Fano 3-folds with terminal singularities

Fano 3-folds with terminal singularities are important in the classification of 3-folds. Fano 3-folds with terminal Gorenstein singularities are roughly classified. However, in the non-Gorenstein case, things get complicated and the classification is not done. In order to make the classification easier, it is useful to consider the deformations. Alt{\i}nok--Brown--Reid conjectured that there exists a deformation of such a Fano 3-fold to that with only quotient singularities. I will talk about my results on this problem.

##### Foliations and Serre-Tate parameters

Serre and Tate defined "canonical coordinates" locally around the moduli point of a polarized ordinary abelian variety. In this talk we will describe two kind of generalizations (joint work with Ching-Li Chai):

-- describe global coordinates;

-- describe a generalization to the non-ordinary case. For the last we need the notion of a foliation. We will describe this, and give properties and examples. Then we describe the proof with the help of a new notion about p-divisible groups over an arbitrary base scheme.

##### Some Results in Complex Hyperbolic Geometry

**PLEASE NOTE SPECIAL DAY AND LOCATION. **In this talk, I present a new approach to the study of cusped complex hyperbolic manifolds through their compactifications. Among other things, I give effective bounds on the number of complex hyperbolic manifolds with given upper bound on the volume. Moreover, I estimate the number of cuspidal ends of such manifolds in terms of their volume. Finally, I address the classification problem for cusped complex hyperbolic surfaces with minimal volume. This is the noncompact or logarithmic analogue of the well known classification problem for fake projective planes.

##### An invariant-theoretic view of the Cox ring and effective cone of \bar{M}_{0,n}

I'll discuss joint work with Brent Doran and Dave Jensen in which we use an "algebraic uniformization" of \bar{M}_{0,n} to study the Cox ring and effective cone. This construction exhibits this moduli space as a non-reductive GIT quotient of affine space and reveals a precise sense in which it is "one G_a away" from being a toric variety. We find, in particular, that for n \ge 7 the Cox ring contains far more information than the effective cone and that several new phenomena arise that do not occur for n \le 6.

##### Rational curves in the log category

In birational geometry, log pairs are introduced for studying open varieties and for reducing problems to lower dimensional case. In this talk, I will explain that this framework can be used to study rational curves on varieties. I will introduce A^1-connectedness for log smooth pairs, i.e., the interior admits lots of rational curves which meet the boundary once. A typical example is a projective space with a smooth Fano hypersurface. Then I will discuss several joint works with Qile Chen as applications of A^1-connected varieties. One application gives a simple proof of general Fano complete intersections are separably rationally connected.

##### Nonarchimedean methods for multiplication maps

Multiplication maps on linear series are among the most basic structures in algebraic geometry, encoding, for instance, the product structure on the graded pieces of the homogeneous coordinate ring of a projective variety. In this talk, I will discuss joint work with Dave Jensen, developing tropical and nonarchimedean analytic methods for studying multiplication maps of linear series on algebraic curves in terms of piecewise linear functions on graphs, with a view toward applications in classical complex algebraic geometry.

##### TBA - Brown

##### On base point freeness in positive characteristic

**Please note special day and time. **Many of the results in the Minimal Model Program depend on Kodaira vanishing theorem and its generalizations. On the other hand, because of the failure of these tools in positive characteristic, many of these results are still open in this case. I will describe some recent progress towards the base point free theorem and the cone theorem over an algebraically closed field of positive characteristic.

##### On the volume of isolated singularities

Boucksom, de Fernex and Favre defined a non-log-canonical volume to study isolated singularities and developed several geometric results. Their definition is based on the intersection number of nef envelope of the log discrepancy b-divisor. In this talk, I will give an alternate definition of this volume by log canonical modification. I will study the lower bound of the volume in Gorenstein case using the new definition. Finally, I will give a threefold X with volume 0 but has no boundary \Delta such that the pair (X,\Delta) is log canonical.

##### The geometric genus of normal surface singularities

We discuss several topological characterizations of the geometric genus of a complex normal surface singularity under certain topological and analytic restrictions. The `classical' cases include the rational and elliptic singularities. More recent characterizations in terms of the Seiberg-Witten invariant and lattice cohomology of the link include more general classes (weighted homogeneous, splice quotients). In terms of the multi-variable divisorial filtration we explain how the Seiberg-Witten invariant appears naturally, and we provide some motivation for the definition of the lattice cohomology as well.

##### Positivity for Weil divisors

We define and study positivity (nefness, amplitude, bigness and pseudo-effectiveness) for Weil divisors on normal projective varieties. We prove various characterizations, vanishing and non-vanishing theorems for cohomology, global generations statements, and a result related to log Fano. Joint work with S. Urbinati.

##### Rota's conjecture and positivity of algebraic cycles in toric varieties

**This is a joint Algebraic Geometry / Discrete Mathematics seminar. ** Rota's conjecture predicts that the coefficients of the characteristic polynomial of a matroid form a log-concave sequence. I will outline a proof for representable matroids using the Bergman fan. The same approach to the conjecture in the general case (for possibly non-realizable matroids) leads to several intriguing questions on higher codimension algebraic cycles in the toric variety associated to the permutohedron.

##### Etale Motovic cohomology and algebraic cycles

**Please note special day and time. **The Chow groups of smooth projective varieties may have torsion and co-torsion which are complicated, except in certain good cases like divisors or 0-cycles. In this talk I will discuss examples of these phenomena, and also variants, considered first by Lichtenbaum, which seem to have better properties in this respect. This is a report on some joint work with A. Rosenschon.

##### Comparing multiplier ideals to test ideals on numerically \mathbb{Q}-Gorenstein varieties

In this talk, I will focus on the connection between two important measures of singularities: multiplier ideals in characteristic zero and test ideals in positive characteristic. While their relationship is well understood in many cases (e.g. hypersurface or finite quotient singularities), it remains conjectural for non-\mathbb{Q}-Gorenstein varieties (such as the cone over the Segre embedding of \mathbb{P}^1 \times \mathbb{P}^2 in \mathbb{P}^5). I will discuss positive recent progress on this conjecture for so-called numerically \mathbb{Q}-Gorenstein varieties (which include all normal surface singularities). This is joint work with T. de Fernex, R. Do Campo, and S. Takagi.

##### Where is the curve in the B-model?

We will describe a rigorous perturbative quantization of the B-twisted topological sigma-model via AKSZ-formalism on derived mapping space around constant maps. We show that the first Chern-class of the target manifold is the obstruction to the quantization, and the factorization algebra of observables on Calabi-Yau target give the expected topological correlation functions in the B-model. This is joint work with Qin Li.

##### Incidences of lines in 3-space

Let L be a set of m distinct lines 3-space. Elekes and Sharir conjectured that, aside from some obvious counter examples, the lines in L have at most C m^{3/2} intersection points. This was proved by Guth and Katz. I will explain the proof with some improvements that make the constant C effective.