# Warwick Algebraic Geometry Seminar

### Summer Term 2020

### Previous terms:

### Spring Term 2020

### Autumn Term 2019

### Summer Term 2019

### Spring Term 2019

### Autumn Term 2018

[[The Warwick Algebraic Geometry Seminar will be taking place this term on Tuesday afternoons at 2pm in MS.05, unless otherwise stated. We also have a later slot available to us on Tuesdays at 4pm in B3.03, which we may make use of occasionally.]]

[[In addition to our own activities, we will also be arranging regular trips to various algebraic geometry activities running in the UK, including the COW seminar, the East Midlands Seminar in Geometry (EmSG), the London Geometry and Topology Seminar, the GLEN seminar, and the British Algebraic Geometry meeting (BrAG).]]

If you are interested in receiving announcements about upcoming seminars and other algebraic geometry activities at Warwick, you're welcome to join our mailing list. To do this, just send an email to Chunyi Li (c.li.25 (at) warwick (dot) ac (dot) uk) and ask to be added to the list.

This term the seminar is online over Zoom. You can register to receive the weekly announcements (including Zoom meeting ID and password) at
**Registration**.

Guidelines for the speaker:

1) Use the option 'Hide non-video participants' in video settings.

2) Share screen and keep your presentation in fullscreen mode all throughout.

3) Share your presentation in advance through the chat (or by email to an organizer).

4) Pause for 5-10 sec at the end of each slide to allow for questions.

5) After around 20 min there will be a break for questions (max 10 min), followed by the rest of the talk.

Guidelines for the participants:

1) Unless you're a host, join without video and generally mute your microphone.

2) When asking a question, unmute your microphone and turn on your video.

3) You can ask also a question directly to a chosen participant using the individual chat function.

4) You can ask also questions to the speaker in the 'Everyone' chat room. That chat is monitored by the chair who will ask the questions to the speaker at appropriate times.

## Abstracts

**Roberto Pignatelli (Trento) - Rigid compact complex surfaces that are not infinitesimally rigid**-
A complex manifold is rigid if every small deformation of its complex structure is trivial. The usual argument for proving the rigidity of a complex manifold is by a well known "standard" cohomological criterium. Morrow and Kodaira posed in 1971 the problem of constructing a rigid manifold that does not satisfy it.

I will present a new criterium for rigidity of a manifold of dimension 2 that is more general than the standard one. As an application, I will produce a family of examples satisfying our criterium and not the classical one, so answering the above question.

This is a joint work with I. Bauer. **Adam Gyenge (Oxford) - Hilbert and Quot schemes of simple surface singularities**- The Hilbert schemes of points on the affine complex plane has the structure of a Nakajima quiver variety. For a finite subgroup G of SL(2, C), I will discuss the construction of the Hilbert scheme of n points on the Kleinian singularity C^2/G as a Nakajima quiver variety for the framed McKay quiver of G with a specific non-generic stability parameter. I will also present a formula for the generating series collecting the Euler numbers of these varieties, a specific case of which was proved recently by Nakajima. Given enough time, I will explain the analogous problem for certain Quot schemes of C^2/G. (Joint work with Alastair Craw, Soren Gammelgaard and Balazs Szendroi).
**Tyler Kelly (Birmingham) - A few aspects of hypergeometric functions in geometry and arithmetic**- Hypergeometric functions are special functions that go back all the way to Euler and come all the way to Mirror Symmetry, Hodge theory, and Gromov-Witten Theory. I plan to give a (very biased) stroll through some of their contexts/computations in algebraic geometry and Hodge theory, and then explain the analogue of hypergeometric functions over finite fields. This will then give us a way to organise point counts on certain varieties over finite fields. This talk will involve joint work with C Doran, A Salerno, S Sperber, U Whitcher, and J Voight.
**Will Donovan (Yau MSC, Tsinghua University) - Windows on the Pfaffian-Grassmannian correspondence**- The Pfaffian-Grassmannian correspondence has been a key example in the development of Homological Projective Duality: it concerns certain pairs of non-birational Calabi-Yau threefolds which share a mirror partner, and can be proved to be derived equivalent. Physically, such an equivalence is associated to B-brane transport along a path in a mirror symmetry moduli space, and is dependent on the homotopy class of that path: I give a mathematical implementation of this dependency, in terms of mutations of an exceptional collection on the relevant Grassmannian. This follows a physical analysis of Hori and Eager-Hori-Knapp-Romo, and builds on work with Addington and Segal.
**Anthony Várilly-Alvarado (Rice University) - Quasi-hyperbolicity via explicit symmetric differentials**- A surface X is algebraically quasi-hyperbolic if it contains finitely many curves of genus 0 or 1. In 2006, Bogomolov and de Oliveira used asymptotic computations to show that sufficiently nodal surfaces of high degree in projective three-space carry symmetric differentials, and they used this to prove quasi-hyperbolicity of these surfaces. We explain how a granular analysis of their ideas, combined with computational tools and insights, yield explicit results for the existence of symmetric differentials, and we show how these results can be used to give constraints on the locus of rational curves on surfaces like the Barth Decic, Buechi's surface, and certain complete intersections of general type, including the surface parametrizing perfect cuboids. This is joint work with Nils Bruin and Jordan Thomas.
**Michel van Garrel (Warwick) - Prelog Chow rings**- In this joint work with Christian Böhning and Hans-Christian Graf von Bothmer, we explore Chow rings in the setting of log geometry, leading to the construction of prelog Chow rings of normal crossings varieties with smooth components. For a strictly semistable degeneration, the prelog Chow ring of the central fiber admits a specialization morphism from the Chow group of the generic fiber. After introducing the definition, I will describe examples illustration the construction.
**Enrica Mazzon (MPI Bonn) - Tropical affine manifolds from mirror symmetry to Berkovich geometry**- Mirror symmetry is a fast-moving research area at the boundary between mathematics and theoretical physics. Originated from observations in string theory, it suggests that certain geometrical objects (complex Calabi-Yau manifolds) should come in pairs, in the sense that each of them has a mirror partner and the two share interesting geometrical properties. In this talk, I will introduce some notions relating mirror symmetry to tropical geometry, inspired by the work of Kontsevich-Soibelman and Gross-Siebert. In particular, I will focus on the construction of a so-called “tropical affine manifold” using methods of non-archimedean geometry, and the guiding example will be the case of K3 surfaces and some hyper-Kähler varieties. This is based on joint work with Morgan Brown and a work in progress with Léonard Pille-Schneider.
**Jean-Louis Colliot-Thélène (CNRS et Université Paris-Saclay) - On the integral Tate conjecture for 1-cycles on the product of a curve and a surface over a finite field**-
Let
*X*be the product of a smooth projective curve*C*and a smooth projective surface*S*over a finite field*F*. Assume the Chow group of zero-cycles on*S*is just**Z**over any algebraically closed field extension of*F*(example : Enriques surface). We give a simple condition on*C*and*S*which ensures that the integral Tate conjecture holds for 1-cycles on*X*. An equivalent formulation is a vanishing result for unramified cohomology of degree 3. This generalizes a result of A. Pirutka (2016). It is a joint work with Federico Scavia (UBC, Vancouver). **Ana-Maria Castravet (Versailles) - Exceptional collections on moduli spaces of pointed stable rational curves**-
I will report on joint work with Jenia Tevelev answering a
question of
Orlov. We prove that
the Grothendieck-Knudsen moduli spaces of pointed stable rational curves
with n markings admit full, exceptional collections which are invariant
under the action of the symmetric group S
_{n}permuting the markings. In particular, a consequence is that the K-group with integer coefficients is a permutation S_{n}-lattice. **Hans-Christian Graf von Bothmer (Hamburg) - Rigid, not infinitesimally rigid surfaces with ample canonical bundle**- It was a long standing problem of Morrow and Kodaira wether there are compact complex manifolds X with Def(X) a non reduced point. The first examples answering this question in the affirmative were given by Bauer and Pignatelli in 2018. As explained by Roberto Pignatelli in this seminar some weeks ago, these are certain surfaces of general type that have nodal canonical models. These canonical models are rigid AND infinitesimally rigid, while their desingularizations are still rigid, but not infinitesimally rigid anymore. One can therefore ask, if this situation is typical for rigid, not infinitesimally rigid surfaces of general type, or if it is possible to have examples with smooth canonical models. We answer this question also in the affirmative by constructing such a surface X via line arrangements and abelian covers. This construction was inspired by Vakil's version of „Murphy’s law in algebraic geometry“. (This is joint work with Christian Böhning and Roberto Pignatelli.)
**Victor Batyrev (Tübingen) - Fine interior of lattice polytopes: MMP and Mirror Symmetry**- The notion "Fine interior" of a lattice polytope P was introduced by Miles Reid in his famous lectures on canonical singularities. A non-degenerated hypersurface in a d-dimensional algebraic torus T is birational to a Calabi-Yau if and only if the Fine interior of its Newton polytope is a single lattice point. This is the starting point for my talk with the ambitious goal to explain my solutions to MMP and Mirror Symmetry for toric hypersurfaces.

## Getting Here

Directions to the university may be found here. Once you're on campus, the Mathematics Institute is located in the Zeeman building; you can download a map of the campus here.

Please note that if you are arriving by public transport, the University of Warwick is not in fact in the town of Warwick, or indeed anywhere near it. Instead, it is located a short distance southwest of Coventry. If you are coming by train the closest stations are Coventry and Leamington Spa.

To get to campus from Coventry station, the fast option is the direct bus 12X, the slower option the buses 11, 11U; all three leave from stand ER3 at the bus hub outside the railway station. At the time of writing, a single ticket from Coventry station to the university costs £2.10 (contactless) or £2.20 (cash); a day ticket is £3.90 (contactless) or £4 (cash); an all day group ticket costs £8 (£5 after 6pm); please note that the buses from Coventry only accept exact change.

To get to campus from Leamington Spa station you should take bus U1, U2, or U17. Please note that these buses do not leave from directly outside the station; instead, the nearest bus stop is just around the corner on Victoria Terrace. A map of the route may be found here. At the time of writing, a single ticket from Leamington Spa station to the university costs £2.75.

This page is maintained by Michel van Garrel. Please email comments and corrections to michel (dot) van (hyphen) garrel (at) warwick (dot) ac (dot) uk.

Many thanks to Alan Thompson and Liana Heuberger for designing and updating this page and for allowing its carbon copy to appear here.