Warwick Algebraic Geometry Seminar

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 E₆ and E₇ 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.