Algebraic Geometry Seminar

Department of Mathematics
Princeton University

Spring 2017 Lectures and events

Place: Fine Hall 322

Time: Tuesday 4:30-5:30

Date Speaker Title
February 7 Sean Keel
University of Texas at Austin
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.
February 14 Kuan-Wen Lai
Brown University
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.
February 21 (Special time/room: 2 pm, Fine 401) Kenneth Ascher
Brown University
Moduli spaces of weighted stable elliptic surfaces
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.
February 28 (Special time/room: 2 pm, Fine 401) Harold Blum
University of Michigan
The Normalized Volume of a Valuation
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.
March 7 Julie Rana
University of Minnesota
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.
March 14 Huai-Liang Chang
Hong Kong University of Science and Technology
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.
March 28 Eric Riedl
University of Illinois at Chicago
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.
April 4 Noah Giansiracusa
Swarthmore College
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.
April 11 Linquan Ma
University of Utah
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.
April 18 Steven Sam
University of Wisconsin, Madison
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.
April 25 Inna Zakharevich
Cornell University
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. In this talk we show how to lift the zeta function to $K$-theory to produce a map of spaces whose $\pi_0$ is the local zeta function and use this map to find interesting elements in higher $K$-groups corresponding to the Grothendieck ring of varieties.

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