Warwick Algebraic Geometry Seminar

Abstracts

Nero Budur (KU Leuven) - Groups and geometry

The representations of the fundamental group of an algebraic variety form important topological invariants connecting group theory with geometry. In this talk, we focus on the varieties of smallest non-trivial dimension: curves. Using quivers and jet schemes, we show that the geometry of the space of representations has constrained singularities. We apply this to show that the number of complex representations of SL_n(ℤ) of dimension at most m grows at most as the square of m, for a fixed n>2.

Noah Arbesfeld (Imperial College) - K-theoretic Donaldson-Thomas theory and the Hilbert scheme of points on a surface

Tautological bundles on Hilbert schemes of points on surfaces often appear in enumerative and physical computations. We explain how to use K-theoretic Donaldson-Thomas theory to produce dualities satisfied by generating functions built from tautological bundles on the Hilbert scheme of points on the complex plane. We then use these dualities to study the Euler characteristics of tautological bundles on Hilbert schemes of points on general surfaces.

Madeline Brandt (UC Berkeley) - Computing Berkovich Skeleta of Curves

Given a smooth curve defined over a valued field, it is a difficult problem to compute the Berkovich skeleton of the curve. In theory, one can find a semistable model for the curve and then find the dual graph of the special fiber, and this will give the skeleton. In practice, these procedures are not algorithmic and finding the model can become difficult. In this talk, we present a method for doing this in the case of superelliptic curves yⁿ=f(x). The solution is combinatorial and involves studying the covering from the curve to ℙ¹, and recovering data about the Berkovich skeleton from the tropicalization of ℙ¹ together with the marked ramification points. Throughout the talk we will study many examples in order to get a feel for the difficulties of this problem and how the procedure is carried out.

Mathieu Florence (Paris Sorbonne) - Lifting representations of smooth profinite groups

This is joint with Charles De Clercq. Let p be a prime number. Given a profinite group G, it is a natural question to ask whether a (continuous) representation G-->GL_n(Z/pZ) lifts to a representation G-->GL_n(Z/p^2Z)- and then further, to higher torsion. For instance, take G=Gal(F_s/F) to be the absolute Galois group of a field F. Let T/F be either an algebraic torus, or an Abelian variety. Then, the natural representation of G on the p-torsion of T(F_s) almost trivially has this property. After discussing basic facts about lifting, we will define the concept of a smooth profinite group. An absolute Galois group, as well as the algebraic fundamental group of a smooth (not necessarily proper) curve over an algebraically closed F, are smooth. We will show that any two-dimensional representation of a smooth profinite group lifts. This generalizes, replacing Z/pZ by an arbitrary field k of characteristic p, and Z_p by the Witt vectors W(k). For p=2 and k=Z/2Z, we prove this up to dimension n=4. We shall discuss the case of arbitrary dimension, which is work in progress- involving much more subtle mechanisms.

Lawrence Barrott (NCTS Taipei) - Logarithmic Chow theory

Chow groups are an important tool in modern algebraic geometry. Recent work in Mirror Symmetry has highlighted the importance of Logarithmic Geometry in understanding moduli spaces and enumerative problems. This talk will use defining logarithmic Chow groups to motivate various logarithmic variants of classical geometric concepts.

Gonçalo Tabuada (MIT) - A noncommutative/topological approach to some celebrated conjectures of Grothendieck, Tate, Voevodsky and Weil

The Grothendieck standard conjectures, the Tate conjecture(s), the Voevodsky nilpotence conjecture, and the Weil conjecture, play a central role in geometry. Notwithstanding the effort of several generations of mathematicians, the proof of these celebrated conjectures remains elusive (the Weil conjecture was proved by Deligne). The aim of this talk is to give an overview of a recent noncommutative/topological approach which led to the proof of the aforementioned important conjectures in several new cases (and to an alternative proof of the Weil conjecture).

Maksym Fedorchuk (Boston College) - Geometry of the associated form morphism

The associated form morphism is an algebraically constructed morphism from the space of smooth degree d hypersurfaces in an n-dimensional projective space to the space of (GIT semistable) degree (n+1)(d-2) hypersurfaces in the dual space. I will discuss a surprising geometric property of this morphism: The fact that it descends to give a locally closed immersion on the levels of GIT quotients, and that in the resulting new compactification of the GIT moduli space of smooth hypersurfaces the discriminant divisor is often contracted (sometimes to a point). This is a joint work with Alexander Isaev. If time permits, I will describe a few explicit examples of this morphism on moduli spaces of points on the projective line, and moduli spaces of plane curves.

Chunyi Li (Warwick) - Stronger Bogomolov-Gieseker type Inequality and stability condition

The classical Bogomolov inequality gives a bound for the second Chern character of slope stable sheaves on smooth projective varieties. The inequality is known to be sharp for some varieties (e.g. Abelian varieties), as well as non-sharp for some others (e.g. the projective plane). Besides Fano and K3 surfaces, it is always difficult to get stronger Bogomolov type inequalities for other surfaces and higher dimensional varieties. I will talk about the method to set up such inequalities via the Bridgeland stability condition. The stronger Bogomolov type inequality has several implications. One upshot will be the existence of stability condition on smooth quintic threefolds. They are the first examples of Calabi-Yau threefolds with trivial fundamental group known to have stability conditions.

Sjoerd Beentjes (Edinburgh) - The crepant resolution conjecture for Donaldson-Thomas invariants

Donaldson-Thomas (DT) invariants are integers that enumerate curves in a given Calabi-Yau 3-fold. Let X be a 3-dimensional Calabi-Yau orbifold, and let Y be a crepant resolution of its coarse moduli space. When X satisfies the Hard Lefschetz condition, i.e., the fibres of the resolution are at most 1-dimensional, the Crepant Resolution Conjecture (CRC) of Bryan-Cadman-Young gives a precise relation between the generating functions of DT invariants of X and Y. I will discuss joint work with John Calabrese and Jørgen Rennemo in which we interpret the relation of the CRC as an equality of rational functions, and prove it using the motivic Hall algebra and Joyce's wall-crossing formula.

Nebojsa Pavic (Sheffield) - Grothendieck groups of isolated quotient singularities

We study Schlichting's K-theory of the Buchweitz-Orlov singularity category for quasi-projective algebraic schemes. Particularly, we show for isolated quotient singularities that the Grothendieck group of its singularity category is finite torsion and that rational Poincare duality is satisfied on the level of Grothendieck groups. We consider also consequences for the resolution of singularities of such quotient singularities, more concretely we prove a conjecture of Bondal and Orlov on the derived category of rational singularities in the case of quotient singularities.

Farbod Shokrieh (Copenhagen) - Measures on graphs and a Kazhdan’s theorem

Classical Kazhdan's theorem for Riemann surfaces describes the limiting behavior of canonical (Arakelov) measures on finite covers in relation to the hyperbolic measure. I will present a generalized version of this theorem for metric graphs. In particular, I will introduce a notion of "hyperbolic measure" in the context of graphs. (Joint work with Chenxi Wu.)

Matteo Varbaro (Genova) - Singularities, Serre conditions and h-vectors

Let R be a standard graded algebra over a field, and denote by H_R(t) its Hilbert series. As it turns out, multiplying H_R(t) by (1-t)^{dim R} yields a polynomial h(t)=h_0+h_1t+h_2t^2+…+h_st^s, known as the h-polynomial of R. It is well known and easy to prove that if R is Cohen-Macaulay h_i is nonnegative for all i. Since being Cohen-Macaulay is equivalent to satisfying Serre condition (S_i) for all i, it is licit to ask if h_i is nonnegative for all i<=r whenever R satisfies (S_r). As it turns out, this is false in general, but true putting some additional assumptions on the singularities of R. In characteristic 0, it is enough that X= Proj R is Du Bois (in particular, if X is smooth we have the desired nonnegativity). In positive characteristic, assuming R is F-pure (equivalently, if the field is perfect, if X=Proj R globally F-pure), lets things work well. In this talk I will speak of the above results and some of their consequences. This is a joint work with Hailong Dao and Linquan Ma.

Greg Smith (Queen’s, Canada) - Sums of Squares on Real Subvarieties

How does one effectively recognize polynomials that are sums of squares on a real projective subvariety? After discussing the larger context for the question, we will focus on new bounds on the number of terms in a sum-of-squares representation for a quadratic form on a real projective subvariety. This talk is based on joint work with G. Blekherman, R. Sinn, and M. Velasco.