Algebra, Geometry, and Number Theory Seminar

Abstracts

Zsolt Patakfalvi - Stable log-varieties are higher dimensional generalizations of (weighted) stable pointed curves. I will present a joint work with Sándor Kovács on proving the projectivity of this moduli space, by showing that certain Hodge-type line bundles are ample on it. I will also mention applications to the subadditivity of logarithmic Kodaira dimension, and to the ampleness of the CM (Chow-Mumford) line bundle.

Noah Giansiracusa - I'll discuss joint work with Patricio Gallardo in which we revisit the Chen-Gibney-Krashen moduli space compactifying configurations of n distinct points in affine space up to translation and homothety. This is a smooth, projective compactification with normal crossings boundary and a nice modular interpretation in terms of blown-up projective spaces. I'll explain how this space can be constructed as a Chow quotient and explore the geometry related to this idea.

Mckenzie West - The existence of rational points is a question applicable to a wide range of mathematical disciplines. There are many examples of surfaces that have local points at every prime yet have no global points. The Brauer-Manin obstruction can provide an explanation of this phenomenon. I will first define the necessary details of the Brauer-Manin obstruction. Second I will present my results on the Brauer-Manin obstruction for a family of cubic surfaces which was first discussed by Birch and Swinnerton-Dyer in the 1970s. Lastly I will provide an overview of my current work on Brauer-Manin computations for K3 surfaces.

Nicola Tarasca - In this talk, I will discuss loci of curves with subcanonical points inside moduli spaces of curves. For instance, the locus of curves of genus 3 with a marked subcanonical point has two components: the locus of hyperelliptic curves with a marked Weierstrass point, and the locus of non-hyperelliptic curves with a marked hyperflex. I will show how to compute the classes of the closures of these codimension-two loci in the moduli space of stable curves of genus 3 with a marked point. Similarly, I will present the class of the closure of the locus of curves of genus four with an even theta characteristic vanishing with order three at a certain point. Finally, I will discuss the geometric consequences of these computations. This is joint work with Dawei Chen.

Daniel Lowengrub - The Riemann Singularity theorem is a classical theorem relating two important objects associated to smooth curves. It expresses the multiplicity of a point on the theta divisor of the curve in terms of the dimension of the fiber over that point of the Abel Jacobi map from the Hilbert scheme to the Jacobian. Sebastian Casalania-Martin and Jesse Kass proved an analog of this for nodal curves and conjectured what the formula should be for general planar curves. In this talk, we will prove a theorem about Segre classes which will allow us to generalize Fulton's proof of the Riemann Singularity theorem to arbitrary planar curves, and thus obtain the conjectured formula.

Alexander Duncan - We consider smooth complete intersections of two quadrics in even-dimensional projective space. Over an algebraically closed field of characteristic not 2, is has long been known that one can find a basis in which both quadratic forms are diagonal. However, in characteristic 2, one can never diagonalize either quadratic form. We present a normal form which applies over an arbitrary field of characteristic 2. Using this normal form we determine the automorphism group of the variety.

David Harvey - I will give an overview of some of the main algorithms that are used to count points on curves over finite fields, and discuss some recent developments.

Benjamin Schmidt - The Hilbert scheme of n points on a smooth projective surface X is a smooth projective variety by a classical result of Fogarty. A natural question about these spaces is what all their ample divisors are. Using derived category techniques by Bayer and Macrì, we describe the nef cones if X has Picard rank 1, irregularity 0 and n is large. Moreover, we compute the nef cones if X is the blow up of the projective plane in 8 general points for any n. This is joint work with Bolognese, Huizenga, Lin, Riedl, Woolf and Zhao originating from the boot camp of the Algebraic Geometry Summer Institute 2015 in Utah.

Olgur Celikbas - Let (R, m) be a local ring and let I be an integrally closed m-primary ideal of R (e.g., I=m). In a recent work with Sean Sather-Wagstaff, we improve on a result of Goto and Hayasaka, and proved that, I has finite Gorenstein dimension if and only if R is Gorenstein. In this talk I will discuss some of the tools used to establish this characterization of Gorenstein rings.

Emanuele Macrì - In the pre-talk I will review the basic theory of Bridgeland stability conditions on derived categories of smooth projective surfaces, and applications to the study of moduli spaces of sheaves. In the main talk, I will present possible extensions of these results to the case of threefolds. This is based on joint work with Bayer, Bertram, Schmidt, Toda, and Stellari.

Kate Thompson - The pretalk will be geared for a more general audience. We will discuss a few select historically-relevant results concerning quadratic forms over the integers in order to highlight the general tools which are commonly used. Expect to hear about the work of both Fields medalists as well as (current) undergraduates.

That the sum of four squares represents all positive integers is a well-known and celebrated result--there even is a formula for the number of such representations (often presented in undergraduate number theory classes). What happens in the number field analogue? Using Siegel's theory of local densities and Hilbert modular forms, we will answer this question in the case of real quadratic number fields. This includes providing explicit (and, on occasion, sharp) bounds on the Eisenstein coefficients of the associated theta series.

Dmitry Zakharov - A symmetric differential on a complex variety is a section of a symmetric power of the cotangent bundle. The existence of non-trivial symmetric differentials is related to the topological properties of the variety, implying that the fundamental group is large in a suitable sense. I will review some recent results on symmetric differentials, and describe a necessary and sufficient condition for a symmetric differential of rank three on a complex surface to be expressible as a product of closed holomorphic 1-forms. Joint work with Federico Buonerba.

Last semester's seminar.