# Seminars & Events for Algebraic Geometry Seminar

##### Elliptic Calabi-Yau 3-folds, Jacobi forms, and derived categories

By physical considerations, Huang, Katz and Klemm conjectured in 2015 that topological string partition functions for elliptic Calabi-Yau 3-folds are governed by certain Jacobi forms. This gives strong structure results for curve counting invariants of elliptic CY 3-folds. I will explain a mathematical approach to prove (part of) the HKK Conjecture. Our method is to construct an involution on the derived category and use wall-crossing techniques. The talk is based on joint work with Georg Oberdieck.

##### Compactification of strata of abelian differentials

Many questions about Riemann surfaces are related to study their flat structures induced from abelian differentials. Loci of abelian differentials with prescribed type of zeros form a natural stratification. The geometry of these strata has interesting properties and applications to moduli of complex curves. In this talk we focus on the question of compactifying the strata of abelian differentials from the viewpoints of algebraic geometry, complex analytic geometry, and flat geometry. In particular, we provide a complete description of the strata compactification over the Deligne-Mumford moduli space of pointed stable curves. The upshot is a global residue condition compatible with a full order on the dual graph of a stable curve. This is joint work with Bainbridge, Gendron, Grushevsky and Moeller, based on arXiv:1604.08834.

##### Convexity in divisor theory

For toric varieties there is a dictionary relating the geometry of divisors to the theory of polytopes. I will discuss how certain aspects of this dictionary can be extended to divisors on arbitrary smooth projective varieties. These results build upon ideas of Khovanskii and Teissier; as in their work, geometric inequalities and convexity theory play an important role. This is joint work with Jian Xiao.

##### Old and new formulas for degeneracy loci

A very old problem asks for the degree of a variety defined by rank conditions on matrices. The story of the modern approach begins in the 1970's, when Kempf and Laksov proved that the degeneracy locus for a map of vector bundles is given by a certain determinant in their Chern classes. Since then, many variations have been studied -- for example, when the vector bundles are equipped with a symplectic or quadratic form, the formulas become Pfaffians. I will describe recent extensions of these results -- beyond determinants and Pfaffians, and beyond ordinary cohomology -- including my joint work with W. Fulton, as well as work of several others.

##### Tropical curve counting in superabundant geometries

I will discuss a general framework using Artin fans -- certain logarithmic algebraic stacks -- in which to understand the relationship between logarithmic stable maps and tropical curve counting. These objects provide a flexible tool to study correspondences between algebraic and tropical curves. In particular, we obtain new realization theorems for tropical curves in superabundant settings. After explaining some remarkably cheap consequences of this setup, I will discuss an application, joint with Yoav Len, to the enumerative geometry of elliptic curves on toric surfaces.

##### Classical invariant theory and birational geometry of moduli spaces

**PLEASE NOTE SPECIAL START TIME: 5:00. **Invariant theory is a study of the invariant subring of a given ring equipped with a linear group action. Describing the invariant subring was one of the central mathematical problems in the 19th century and many great algebraists such as Cayley, Clebsch, Hilbert, and Weyl had contributed to it. There are many interesting connections between invariant theory and modern birational geometry of moduli spaces. In this talk I will explain some concrete examples including the moduli space of parabolic vector bundles on the projective line and the moduli space of stable rational pointed curves. This talk is based on joint work with Swinarski and Yoo.

##### On Noether's inequality for stable log surfaces

In this talk I report on some recent progress on the geography problem of stable log surfaces. This is about restrictions on their holomorphic invariants, such as the volume K^2 and the geometric genus p_g. Compared to the case of surfaces of general type, a new feature here is that the volume of a stable log surface is not necessarily an integer. Extending the work of Tsunoda and Zhang in the nineties, I will give an optimal lower bound of the volume when the geometric genus is one. Then I will use an example to illustrate that a speculated Noether type inequality for stable log surfaces does not hold in general.

##### Log geometric techniques for open invariants in mirror symmetry

**This is a joint Algebraic Geometry and Symplectic Geometry seminar. Please note different room (322) and start time (4:30). **We would like to discuss an algebraic-geometric approach to some open invariants arising naturally on the A-model side of mirror symmetry. The talk will start with a smooth overview of the use of logarithmic geometry in the Gross-Siebert program. We then will discuss various illustrations of the use in open invariants, including a description of the symplectic Fukaya category via certain stable logarithmic curves. For this, our main object of study will be the degeneration of elliptic curves, namely the Tate curve. However, the results are expected to generalise to higher dimensional Calabi-Yau manifolds.

##### Theta functions for affine log CY varieties

Gross, Hacking, Siebert and I conjecture that the vector space of regular functions on an affine log CY (with maximal boundary) comes with a canonical basis, generalizing the monomial basis on a torus, in which the structure constants for the multiplication rule are given by counts of rational curves on the mirror. Instances are a basis of the Cox ring of a Fano canonically determined by a single choice of anti-canonical divisor, one example of which gives a canonical basis for every irreducible representation of a semi-simple group (without doing any representation theory!). I'll explain the conjecture, these applications, and then some of the ideas in my recent construction, joint with Tony Yu, of the algebra in dimension two using some simple ideas from Berkovich analytic geometry.

##### Cremona Transformations and Derived Equivalences of K3 Surfaces

Two varieties are called derived equivalent if their bounded derived categories of coherent sheaves are isomorphic to each other. In the case of K3 surfaces, this equivalence is realized as an Hodge isometry between the transcendental lattices according to Mukai and Orlov. Could it be realized further through an explicit construction of birational geometry? In this talk, I will present an example where the derived equivalences of K3 surfaces are explained through Cremona transformations of P^4. This example also provides an interesting relation in the Grothendieck ring of complex algebraic varieties. This is joint work with Brendan Hassett.

##### Moduli spaces of weighted stable elliptic surfaces

**Please note special time and location. **I will discuss recent work (with Dori Bejleri) towards constructing various modular compactifications of spaces of elliptic surface pairs analogous to Hassett's moduli spaces of weighted stable curves.

##### The Normalized Volume of a Valuation

**Please note special time and location.** Motivated by work in Kahler-Einstein geometry, Chi Li defined the normalized volume function on the space of valuations over a singularity and proposed the problem of both finding and studying the minimizer of this function. While Li's problem is closely connected to the notion of K-semistability, it also relates to an invariant of singularities previously explored in the work of de Fernex, Ein, and Mustata. I will explain the motivation for this problem and discuss a recent result proving the existence of normalized volume minimizers.

##### The Craighero-Gattazzo surface is simply-connected

We show that the Craighero-Gattazzo surface, the minimal resolution of an explicit complex quintic surface with four elliptic singularities, is simply-connected. This was first conjectured by Dolgachev and Werner, who proved that its fundamental group has trivial profinite completion. This makes the Craighero-Gattazzo surface the only explicitly known example of a smooth simply-connected complex surface of geometric genus zero with ample canonical class. The proof utilizes an interesting technique: to prove a topological fact about a complex surface we use algebraic reduction mod p and deformation theory.

##### All genus Gromov Witten invariant of quintic via Mixed Spin P field

Adding higher obstructions (P fields) into moduli spaces of maps, one represent Gromov Witten invariants of quintic hypersurfaces as Landau Ginzburg type invariants. By promoting Kahler parameter into fields on worldsheet, one obtains a moduli space connecting Gromov Witten invariants with Fan-Jarvis-Ruan-Witten invariants. This moduli is called Mixed Spin P (MSP) fields. We then can approach structures of all genus quintic GW invariants, such as holomorphic ambiguity, CY-LG correspondence, and algorithms.

##### Rational curves in projective space with fixed normal bundle

Given a fixed vector bundle E on P^1, one can ask: what is the moduli space of rational curves in P^n with normal bundle E? For projective 3-space, well-known results of Ghione-Sacchiero and Eisenbud-Van de Ven prove that the space of curves with given normal bundle in P^3 is irreducible of the expected dimension, and Eisenbud and Van de Ven conjecture that the same thing holds for arbitrary P^n. Alzati and Re found a single counterexample to this conjecture in P^8. In this talk, I describe joint work with Izzet Coskun finding an infinite family of counterexamples to the conjecture, where we show that the moduli spaces of rational curves with fixed normal bundle can have arbitrarily many components.

##### Tropical Schemes

Tropical scheme theory is a method of describing tropical varieties with equations, in order to incorporate more foundations and constructions from modern algebraic geometry into the subject. I'll give an overview of this topic, emphasizing recent connections to matroid theory.

##### Lech's conjecture

A long-standing conjecture of Lech states that the Hilbert-Samuel multiplicity does not drop for faithfully flat extensions of local rings. This conjecture was known in dimension less than or equal to two and remains open in higher dimensions. In this talk we will use Hilbert-Kunz theory to obtain estimates on the multiplicities under faithfully flat extension, and in particular we can prove the conjecture in dimension three, in equal characteristic.

##### Secant varieties of Veronese embeddings

Given a projective variety X over a field of characteristic 0, and a positive integer r, we study the rth secant variety of Veronese re-embeddings of X. In particular, I'll explain recent work which shows that the degrees of the minimal equations (and more generally, syzygies) defining these secant varieties can be bounded in terms of X and r independent of the Veronese embedding. This is based on arXiv:1510.04904 and arXiv:1608.01722.

##### Constructing a derived zeta function

The local zeta function of a variety $X$ over a finite field $k$ can be defined to be $\sum_{n \geq 1} |(\mathrm{Sym}^nX)(k)| t^n$. As this depends only on the point counts of symmetric powers of $X$ it is an invariant of the class of $X$ in the Grothendieck ring of varieties $K_0(\mathrm{Var}_k)$: the ring which is generated by varieties over $k$ modulo the relation that whenever $Z$ is a closed subvariety of $Y$ we have $[Y] = [Z] + [Y\backslash Z]$. In fact, the local zeta function can be thought of as having codomain in the big WItt ring. Both the Grothendieck ring of varieties and the Witt ring appear as the $0$-th $K$-theory groups of certain categories. |