# Warwick Algebraic Geometry Seminar

### Spring Term 2020

### Previous terms:

### 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.

## Abstracts

**Miles Reid (University of Warwick) - G-Hilb for trihedral groups****Livia Campo (University of Warwick) - Index 2 Fano 3-folds and their birational geometry**- In this talk I will construct, by means of unprojection techniques, 34 of the 37 Fano 3-folds in codimension 4 and index 2 whose Hilbert series are listed in the Graded Ring Database. If time permits, I will also discuss their birational non-rigidity by studying their associated Sarkisov links (joint work with Tiago Guerreiro), comparing this result with the index 1 case.
**Timothy Magee (University of Birmingham) - Convexity in tropical spaces and compactifications of cluster varieties**- Cluster varieties are a relatively new, broadly interesting class of geometric objects that generalize toric varieties. Convexity is a key notion in toric geometry. For instance, projective toric varieties are defined by convex lattice polytopes. In this talk, I'll explain how convexity generalizes to the cluster world, where "polytopes" live in a tropical space rather than a vector space and "convex polytopes" define projective compactifiactions of cluster varieties. Time permitting, I'll conclude with two exciting applications of this more general notion of convexity: 1) an intrisic version of Newton-Okounkov bodies and 2) a possible cluster version of a classic toric mirror symmetry construction due to Batyrev. Based on joint work with Man-Wai Cheung and Alfredo Nájera Chávez.
**Evangelos Routis (Warwick) - Complete Complexes and Spectral Sequences**- The space of complete collineations is an important and beautiful chapter of algebraic geometry, which has its origins in the classical works of Chasles, Schubert and many others, dating back to the 19th century. It provides a 'wonderful compactification' (i.e. smooth with normal crossings boundary) of the space of full-rank maps between two (fixed) vector spaces. More recently, the space of complete collineations has been studied intensively and has been used to derive groundbreaking results in diverse areas of mathematics. One such striking example is L. Lafforgue's compactification of the stack of Drinfeld's shtukas, which he subsequently used to prove the Langlands correspondence for the general linear group. In joint work with M. Kapranov, we look at these classical spaces from a modern perspective: a complete collineation is simply a spectral sequence of two-term complexes of vector spaces. We develop a theory involving more full-fledged (simply graded) spectral sequences with arbitrarily many terms. We prove that the set of such spectral sequences has the structure of a smooth projective variety, the 'variety of complete complexes', which provides a desingularization, with normal crossings boundary, of the 'Buchsbaum-Eisenbud variety of complexes', i.e. a 'wonderful compactification' of the union of its maximal strata.
**Anne-Sophie Kaloghiros (Brunel) - Volume preserving maps of Calabi-Yau pairs with a toric model**-
A Calabi-Yau pair (X, D) consists of a normal projective variety and a reduced integral divisor D with K
_{X}+ D ~ 0. A birational map (X, D_{X}) - - > (Y, D_{Y}) between CY pairs is volume preserving if pullbacks of K_{X}+ D_{X}and K_{Y}+ D_{Y}to high enough models coincide. The pair formed by a toric variety and its boundary divisor is an example of CY pair, and mutations of algebraic tori (C^{∗})^{n}- - >(C^{∗})^{n}can be extended to volume preserving birational maps of toric pairs. In this talk, I will discuss a conjecture that states that every volume preserving birational map between CY pairs that have a toric model (ie that are volume preserving birational to toric pairs) is a chain of mutations. I will give some evidence in support of the conjecture in dimension 3. **Andrea Cattaneo (Florence) - Towards a Lefschetz-type phenomenon for elliptic Calabi-Yau manifolds**-
Many contractions are known to produce elliptically fibered
manifolds: the easiest examples are smooth Weierstrass fibrations,
then there are some "classical" families known in the literature as
E
_{6}and E_{7}families, and other constructions such as the Borcea-Voisin construction. Depending on some choice involved in the construction, the resulting elliptic manifold may have further properties, e.g., it can be a Calabi-Yau manifold. As it is known from general theory, all these examples can be obtained also from a singular Weierstrass fibration by means of a resolution of the singularities. The aim of our talk is to provide evidences of the following (up to now) conjectural phenomenon: under some assumptions on the resolution of the singularities, the Hodge diamond of the resolved manifold and those of its blown up ambient space coincide in the degrees predicted by the Lefschetz Hyperplane Theorem, despite the fact that our manifolds are not ample divisors in their ambient spaces. We will also stress the improtance of the Calabi-Yau condition, providing examples of non Calabi-Yau elliptic fibrations which does not show this phenomenon. This is joint work with James Fullwood (SJTU). **Antonella Grassi (Bologna) - Zariski decomposition for higher dimensional elliptic fibrations**- A sketch of the ideas of the joint preprint arXiv 1904.02779 with David Wen, intended to stimulate discussion.
**Marta Panizzut (TU Berlin) - Moduli space of tropical cubic surfaces**- Recent works of Ren, Sam, Shaw and Sturmfels focused on tropicalizing classical moduli spaces from their defining equations. A particularly interesting example is the moduli space of marked del Pezzo surfaces of degree three. Its tropical version is the 4-dimensional Naruki fan and its maximal cones reveal two generic types of tropical cubic surfaces characterized by their structure at infinity, which is an arrangement of 27 trees with 10 leaves. In this talk we begin by illustrating this construction. We will then introduce an octanomial model for cubic surfaces. This new normal form is well suited for p-adic geometry, as it reveals the intrinsic del Pezzo combinatorics of the 27 trees in the tropicalization. The talk is based on joint work with Emre Sertöz and Bernd Sturmfels.
**Patricio Gallardo (Washington University in St. Louis) - Geometric interpretations of Hodge theoretical compactifications of ball quotients**- It is known, due to the work of Deligne and Mostow, that some GIT compactifications of moduli spaces of points in the projective line are isomorphic to the Baily-Borel compactification of an appropriate ball quotient. In ongoing work with L. Schaffler and M. Kerr, we show that for each Deligne-Mostow ball quotient, the corresponding unique toroidal compactification is isomorphic to a Hasset's moduli space of weighted stable rational curves. Similar results relating to the KSBA and toroidal compactifications of the moduli space of cubic surfaces will be discussed.

## 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.