Talks

All times are in GMT+1 (the current time in the UK) and all talks are in-person.

For a list of talks ordered by date and time, see the Schedule page.

19th July, 2022

Jesus Martinez-Garcia (Essex)- Some observations on the dimension of Fano K-moduli

10:00-11:00

Abstract: Moduli spaces of K-stable Fano varieties have been intensively investigated in the last decade, both from a general theory point of view as well as via the study of explicit examples, mostly on the smoothable setting, i.e. where the general member of the moduli is smooth. However, the theory has now been extended to the non-smoothable setting, where a number of oddities can appear. In this talk, we will demonstrate via examples, how for any n>1, the dimension (as a scheme) of the K-moduli stack of n-dimensional Fano varieties is unbounded. We will also show how the dimension of the K-moduli stack can be arbitrarily big, while the dimension of its coarse variety remains bounded. This is joint work with Cristiano Spotti.

Marta Benozzo (Imperial)- Iitaka Conjecture for Anticanonical divisors in Positive Characteristic

11:30-12:30

Abstract: Given a fibration f: X -> Z, a natural question is how we can relate the Kodaira dimensions of X, Z, and the fibres. The Iitaka conjecture addresses this problem. In characteristic 0, it is proven in many cases, while it is known to not hold in general in positive characteristics. Recently, a similar statement for the anticanonical divisors was proven in characteristic 0. The same results can be extended also to positive characteristics in low dimensions. At the end, we will see also some counterexamples constructed from Tango-Raynaud surfaces.

Calum Spicer (Imperial)- Moduli of foliated surfaces

14:00-15:00

Abstract: A foliation on an algebraic surface is a partition of the surface into disjoint immersed holomorphic submanifolds. Foliations arise naturally in a wide range of contexts in geometry, for instance in the study of rational curves on varieties or in the study of hyperbolicity properties of varieties. In this talk I will explain some recent work related to constructing moduli spaces of foliations.

There are several unique properties of foliation geometry which pose special difficulties when attempting to construct these spaces.

I will explain how to address these difficulties using some recent developments in the Minimal Model Program for foliations. This is joint work with Roberto Svaldi.

Antonio Trusiani (Toulouse)- The Yau-Tian-Donaldson conjecture for a relative K-stability notion

15:15-16:15

Abstract: In the first part of the talk, a new relative K-stability notion will be defined. A particular focus will be given to the motivations and intuitions, making a comparison with the log K-stability. Then a Yau-Tian-Donaldson correspondence will be presented, relating this algebro-geometric notion to the existence of a class of special metrics, called Kähler-Einstein metrics with prescribed singularities. Time permitting, a discussion on some future developments concerning a relative δ-invariant will conclude the talk.

20th July, 2022

Artie Prendergast-Smith (Loughborough) - Birational Geometry of mixed products of projective spaces

10:00-11:00

Abstract: We examine some aspects of the birational geometry of blowups of products P^1 x P^2 and P^1 x P^3 in small numbers of points. In particular we explain how to compute their effective cones, and show that they are Mori dream spaces by exhibiting log Fano structures on them. Joint with Tim Grange and Elisa Postinghel.

Tom Ducat (Durham) - Log Calabi-Yau 3-folds with tropicalisation R^3/G

11:30-12:30

Abstract: The tropicalisation of a log Calabi--Yau variety U is an integral affine manifold with singularities (IAMS) which is a generalisation of the cocharacter space when U is a torus. The quotient spaces R^2/G, for G one of the four nontrivial finite subgroups of SL(2,Z), are all 2-dimensional IAMS which appear naturally in the classification of tropicalisations of log Calabi--Yau surfaces due to Mandel. For the 23 nontrivial finite subgroups G of SL(3,Z) I will explain how to construct a family of log Calabi--Yau 3-folds whose tropicalisation is R^3/G.

Tom Hall (Nottingham)- On the uniqueness Kähler-Einstein polygons in mutation-equivalence classes

14:00-15:00

Abstract: An important object of study in the programme of mirror symmetry is that of the Fano polytope. Classifying these up to mutation conjecturally gives a classification of Fano varieties. We investigate the behaviour of Kähler-Einstein polygons, i.e. Fano polygons which correspond to toric Fano varieties admitting a Kähler-Einstein metric. In particular, we focus on the subcase of centrally symmetric Fano polygons and show that there is at most one per mutation-equivalence class.

Joshua Jackson (Imperial)- Non-reductive GIT and moduli of hypersurfaces

15:15-16:15

Abstract: I'll explain how to use new results in non-reductive GIT to construct moduli spaces for hypersurfaces in weighted projective spaces (or toric orbifolds), generalising Mumford's construction for ordinary projective space. This is joint work with Dominic Bunnett.

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