Quasilinear tropical compactifications

A tropical compactification of a subvariety of a torus is a closure of the subvariety in some toric variety, satisfying some basic properties. I will describe ongoing work introducing and studying the class of quasilinear tropical compactifications, which generalize compactifications of complements of hyperplane arrangements. Quasilinear tropical compactifications satisfy a number of desirable properties; for instance, their Chow and cohomology rings are the same as the Chow and cohomology rings of the ambient toric variety. This theory leads to an understanding of some interesting compactifications of moduli spaces, in particular moduli of hyperplane arrangements and marked del Pezzo surfaces.

Compact modul of K3 surfaces with a non-symplectic automorphism

We construct a functorial, geometrically meaningful compactification of the moduli spaces of K3s in the title, and prove that it is semi-toroidal. This is joint work with Phil Engel and Changho Han.

Serre functors of semiorthogonal components

The Serre functor of a triangulated category is one of its most important invariants, playing the role of the dualizing complex of a variety in noncommutative algebraic geometry. I will explain how to describe the Serre functors of many semiorthogonal components of varieties in terms of spherical twists. In the case of Kuznetsov components of Fano complete intersections, this leads to a proof of a conjecture of Katzarkov and Kontsevich on the dimensions of such categories, and implies the nonexistence of Serre invariant stability conditions when the degrees of the complete intersection do not all coincide. This is joint work with Alexander Kuznetsov.

Cayley-Bacharach theorems and measures of irrationality

If Z is a set of points in projective space, we can ask which polynomials of degree d vanish at every point in Z. If P is one point of Z, the vanishing of a polynomial at P imposes one linear condition on the coefficients. Thus, the vanishing of a polynomial on all of Z imposes |Z| linear conditions on the coefficients. A classical question in algebraic geometry, dating back to at least the 4th century, is how many of those linear conditions are independent? For instance, if we look at the space of lines through three collinear points in the plane, the unique line through two of the points is exactly the one through all three; i.e. the conditions imposed by any two of the points imply those of the third. In this talk, I will survey several classical results including the original Cayley-Bacharach Theorem and Castelnuovo’s Lemma about points on rational curves. I’ll then describe some recent results and conjectures about points satisfying the so-called Cayley-Bacharach condition and show how they connect to several seemingly unrelated questions in contemporary algebraic geometry relating to the gonality of curves and measures of irrationality of higher dimensional varieties.

K-stability and birational geometry of moduli spaces of quartic K3 surfaces

I will discuss various compactifications of moduli spaces of quartic K3 surfaces, coming from geometric invariant theory (GIT), Hodge theory, and K-stability. We will see that K-stability provides a natural interpolation between various other compactifications via wall crossings in K-moduli and prove conjectures of Laza and O'Grady along with new results. This is joint work with Kenneth Ascher and Yuchen Liu.

An overview of Non-Reductive Geometric Invariant Theory and its applications

Geometric Invariant Theory (GIT) is a powerful theory for constructing and studying the geometry of moduli spaces in algebraic geometry. In this talk I will give an overview of a recent generalisation of GIT called Non-Reductive GIT, and explain how it can be used to construct and study the geometry of new moduli spaces. These include moduli spaces of unstable objects (for example unstable Higgs/vector bundles), hypersurfaces in weighted projective space, k-jets of curves in C^n and curve singularities.

Real fibered morphisms of real del Pezzo surfaces

A morphism of smooth varieties of the same dimension is called real fibered if the inverse image of the real part of the target is the real part of the source. It goes back to Ahlfors that a real algebraic curve admits a real fibered morphism to the projective line if and only if the real part of the curve disconnects its complex part. Inspired by this result, in a joint work with Mario Kummer and Cédric Le Texier, we are interested in characterising real algebraic varieties of dimension n admitting real fibered morphisms to the n-dimensional projective space. We present a criterion to construct real fibered morphisms that arise as finite surjective linear projections from an embedded variety; this criterion relies on topological linking numbers. We address special attention to real algebraic surfaces. We classify all real fibered morphisms from real del Pezzo surfaces to the projective plane and determine when such morphisms arise as the composition of a projective embedding with a linear projection.

A non-hypergeometric E-function

In a landmark 1929 paper, Siegel introduced the class of E-functions with the goal of generalising the transcendence theorems for the values of the exponential. E-functions are power series with algebraic coefficients subject to certain growth conditions that satisfy a linear differential equation. Besides the exponential, examples include Bessel functions and a rich family of hypergeometric series. Siegel asked whether all E-functions are polynomial expressions in these hypergeometric series. I will explain why the answer is negative (joint work with Peter Jossen).

Isogenous hyper-Kähler varieties

The Torelli theorem for hyper-Kähler varieties explains to which extent such a variety can be recovered from its integral second cohomology group, together with its pairing and Hodge structure. In my talk I will address a variant of this question: how much of a hyper-Kähler variety is encoded in its rational second cohomology group? For K3 surfaces the answer is provided by work of Huybrechts and Fu-Vial. In higher dimension we expect this rational cohomology group to control the full motive of the variety. I will explain how this can be made precise in the realm of André motives.