# Durham University

## 11-12 September 2024

Our next conference will take place at Durham University on Wednesday 11th and Thursday 12th September 2024.

All talks will take place in room MCS 2068, on the second floor of the Mathematical and Computer Sciences Building. A map with the location can be found towards the bottom of the following page:

Schedule

The schedule for the event will be as follows; abstracts may be found below.

Wednesday 11th September

1:30pm - Daniele Turchetti (Durham) - Degenerations of varieties through the lens of non-Archimedean geometry

2:30pm - Tea/coffee break

3:00pm - Holly Krieger (Cambridge) - Dynamical degrees of birational maps

4:30pm - Alessio Corti (Imperial) - Birational geometry of log structures

Thursday 12th September

9:15am - Thibault Poiret (St Andrews) - Spaces of roots of universal line bundles

10:15am - Tea/coffee break

10:45am - Arman Sarikyan (LIMS) - On the Rationality of Fano-Enriques Threefolds

12:00pm - Simon Felten (Oxford) - Global logarithmic deformation theory

Accommodation and Travel

We will provide overnight accommodation on 11th September for a limited number of UK-based students and postdocs. We also have some funding to cover travel costs for UK-based students and postdocs. To ensure that we can fund as many participants as possible, we ask that participants purchase "advance" or "off-peak" train tickets where practical. For those under the age of 30, we also recommend looking into getting a railcard which can offer substantial savings on the cost of train travel around the UK.

If you would like to apply for travel funding, please indicate this on the registration form. The deadline for applying for accommodation has now passed. In case we receive more applications than we have funding to cover, travel funding will be allocated on a first-come, first-served basis.

Abstracts

Daniele Turchetti (Durham) - Degenerations of varieties through the lens of non-Archimedean geometry

Degeneration techniques have been classically employed to solve problems in algebraic geometry for many years. More recently, connections between polyhedral and tropical geometry have been uncovered by several authors (Nicaise—Xu and Gubler—Rabinoff—Werner among others), thanks to the use of non-Archimedean analytic geometry in the sense of Berkovich. More precisely, one can associate with certain models of a smooth project variety X over a non-Archimedean field a polyhedral complex, called the skeleton, that naturally lives inside the Berkovich analytification of X and that encodes crucial properties of the corresponding degeneration.

In this talk, I will survey this non-archimedean analytic perspective on degenerations of algebraic varieties and provide the audience with specific situations where Berkovich’s theory has been used to solve problems, including joint work with Lorenzo Fantini and Andrew Obus on models of curves over DVRs and joint work in progress with Christian Boehning on degenerations of projective spaces and toric varieties.Holly Krieger (Cambridge) - Dynamical degrees of birational maps

In the study of the discrete dynamical system defined by a rational self-map of a projective variety, we hope as a starting point to understand the growth of the degrees of the iterates of the map. This growth is measured by the dynamical degree, an invariant which controls the topological, arithmetic, and algebraic complexity of the system. I will discuss the history of this question and the recent surprising construction, joint with Bell, Diller, and Jonsson, of a transcendental dynamical degree for a birational map, and how our work fits into the general phenomenon of power series taking transcendental values at algebraic inputs.Alessio Corti (Imperial) - Birational geometry of log structures

There are many reasons for wanting to work with singular log structures, and for specifying classes of mildly singular log structures. I show some examples of log singular log structures that admit interesting log crepant partial log resolutions. These examples suggest some conjectures and hint at a possible general theory. I may sketch some applications to mirror symmetry and the theory of minimal models. This is work in progress with Tim Graefnitz and Helge Ruddat.Thibault Poiret (St Andrews) - Spaces of roots of universal line bundles

Over the moduli space Mg,n of smooth curves, the only relative line bundles on the universal curve are linear combinations of the cotangent bundle ω and of the classes of the marked sections xi. Let L=ωk(Σi mi xi) be such a linear combination. The space S1/r(L) of r-th roots of L forms a finite cover of Mg,n. When r=2 and L=ω, this is the space of so-called spin curves.

One can find meaningful compactifications of S1/r(L), finite over the Deligne-Mumford-Knudsen compactification of Mg,n parametrizing stable curves. I will talk about some aspects of the geometry of these compactifications.Arman Sarikyan (LIMS) - On the Rationality of Fano-Enriques Threefolds

A three-dimensional non-Gorenstein Fano variety with at most canonical singularities is called a Fano-Enriques threefold if it contains an ample Cartier divisor that is an Enriques surface with at most canonical singularities. Such threefolds always admit a double covering by Gorenstein Fano varieties, similarly to how Enriques surfaces always admit a double covering by a K3 surface. There is no complete classification of Fano-Enriques threefolds yet, although some partial results are known. In this talk we will discuss geometry of Fano-Enriques threefolds, their birational properties, and will study their rationality.Simon Felten (Oxford) - Global logarithmic deformation theory

Funding

This event is taking place as part of the UK Algebraic Geometry Network. The network is funded by a Network Support for the Mathematical Sciences grant from the Isaac Newton Institute, via EPSRC grant EP/V521929/1.