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Algebraic Geometry Seminar 21/22 Term 3

The algebraic geometry seminar in Term 3 2021/22 will usually meet on Wednesdays at 3pm, though we will sometimes change to allow speakers from other time zones.

See the talks from last term here.

Wednesday 11 May 2022, 3pm. Speaker: Siddarth Kannan (Brown) (in-person talk)

Title: Moduli of relative stable maps to P^1: cut-and-paste invariants

Abstract: I will give an introduction to the moduli space of genus zero rubber stable maps to P^1, relative to 0 and infinity, with fixed ramification profiles. Then I will discuss two recent results on the topology of these moduli spaces. The first concerns a chamber structure for the classes of these moduli spaces in the Grothendieck ring of varieties. The second gives a recursive algorithm for the calculation of the Euler characteristic, in the case where the maps are fully ramified over zero, and unramified over infinity. If time permits, I will also discuss some potential future directions.

Wednesday 18 May 2022, 3pm. Speaker: Eric Jovinelly (Notre Dame) (virtual talk)

Title: Extreme Divisors on M_{0,7} and Differences over Characteristic 2

Abstract: The cone of effective divisors controls the rational maps from a variety. We study this important object for M_{0,n}, the moduli space of stable rational curves with n markings. Fulton once conjectured the effective cones for each n would follow a certain combinatorial pattern. However, this pattern holds true only for n < 6. Despite many subsequent attempts to describe the effective cones for all n, we still lack even a conjectural description. We study the simplest open case, n=7, and identify the first known difference between characteristic 0 and characteristic p. Although a full description of the effective cone for n=7 remains open, our methods allowed us to compute the entire effective cones of spaces associated with other stability conditions.

Wednesday 1 June 2022, 3pm. Speaker: Andreas Hoering (Nice)

Title: Fano manifolds with big tangent bundle

Abstract: Let X be a Fano manifold, that is a smooth projective manifold such that the anticanonical bundle det T_X is ample. On the one hand it is well known that the positivity of the anticanonical bundle rarely implies a positivity property for the tangent bundle T_X (for example Mori's theorem tells us that T_X is ample only when X is the projective space). On the other hand, manifolds with a "positive" tangent have a very rich geometry and are therefore particularly interesting. In this talk I will discuss the case of manifolds with big tangent bundle: we will see that there are many pathologies, but that the study of the rational curves makes it possible to show some first classification results. This talk is based on a joint work with Jie Liu.

Wednesday 15 June 2022, 3pm. Speaker: Johannes Nicaise (Imperial/Leuven)

Title: Variation of stable birational type and bounds for complete intersections

Abstract: This talk is based on joint work with John Christian Ottem. I will first recall the construction of a specialization map for stable birational types (joint work with Evgeny Shinder). Next, I will show how it can be used to extend Schreieder's non-stable rationality bounds from hypersurfaces to complete intersections in characteristic zero. This technique yields new results already for complete intersections of quadrics.

Wednesday 22 June 2022, 3pm. Speaker: Jerome Poineau (Caen Normandie)

Title: Torsion points of elliptic curves via Berkovich spaces over Z

Abstract: Berkovich spaces over Z may be seen as fibrations containing complex analytic spaces as well as p-adic analytic spaces, for every prime number p. We will give an introduction to those spaces and explain how they may be used in an arithmetic context to prove height inequalities. As an application, following a strategy by DeMarco-Krieger-Ye, we will give a proof of a conjecture of Bogomolov-Fu-Tschinkel on uniform bounds on the number of common images on P^1 of torsion points of two elliptic curves.

Wednesday 29 June 2022, 3pm. Speaker: Xuanchun Lu (Warwick)

Title: Constructing Mirrors for Log Calabi-Yau Surfaces

Abstract: This is an expository talk explaining the mirror constructions by Gross, Hacking and Keel in [GHK16]. Following the slogan "mirror symmetry is combinatorial duality", we will explain how the mirror family to a log Calabi--Yau surface is constructed from the combinatorial information of a polarised tropical manifold B and a consistent scattering diagram D. By counting certain piecewise-linear maps to B, we will then be able to describe a basis of regular functions on the mirror and their relations, thereby giving an explicit embedding in the positive case. Finally, we will work through an extended example applying the construction to the del Pezzo surface of degree 5.