Warwick Algebraic Geometry Seminar

Abstracts

Isac Heden (Warwick) - The Borel subgroups of the Cremona group

The defining property of a Borel subgroup in a linear algebraic group is that it is maximal among closed connected solvable subgroups. It is well known that all Borel subgroups are conjugate in an algebraic group. We investigate the situation in the case of the Cremona group. This is a joint work with Jean-Philippe Furter.

Miles Reid (Warwick) - TOp II

We construct Godeaux surfaces and Calabi-Yau 3-folds whose contains 5-torsion, in mixed characteristic at 5. The same ideas give Campedelli surfaces and Calabi-Yau 3-folds with 7-torsion, and (eventually) Godeaux surfaces with 3-torsion. The aim is to put varieties in characteristic p with action of ℤ/p, α_p and μ_p into a single family, together with the characteristic zero case with a ℤ/p action.

Honglu Fan (ETH Zurich) - Absolute and relative quantum cohomology

In a joint work with Longting Wu and Fenglong You, we defined relative quantum cohomology rings by introducing relative Gromov-Witten invariants with negative contact orders. I will compare the definitions of absolute and relative quantum rings, try to explain how the WDVV equation (associativity) motivates our definition of negative contacts, and perhaps show some examples if time permits.

Pierrick Bousseau (ETH Zurich) - Curves, quivers, and Calabi-Yau 4-folds

I will discuss various surprising connections in enumerative geometry. The starting point will be the question of counting rational curves in the projective plane, passing through a given point, and maximally tangent to a given line and to a given conic. From there, we will establish connection with a Calabi-Yau 4-fold, a quiver, and a dual Calabi-Yau 3-fold. Based on work in progress with Andrea Brini and Michel van Garrel.

Tom Ducat (Imperial) - A Laurent phenomenon for OGr(5,10) and explicit mirror symmetry for the Fano 3-fold V_12

The 5-periodic birational map (x, y) -> (y, (1+y)/x) can be interpreted as a mutation between five open torus charts in a del Pezzo surface of degree 5, coming from a cluster algebra structure on the Grassmannian Gr(2,5). This can used to construct a rational elliptic fibration which is the Landau--Ginzburg mirror to dP_5. I will briefly recap all of this, and then explain the following 3-dimensional generalisation: the 8-periodic birational map (x, y, z) -> (y, z, (1+y+z)/x) can be used to exhibit a Laurent phenomenon for the orthogonal Grassmannian OGr(5,10) and construct an explicit K3 fibration which is mirror to the Fano 3-fold V_12.

Michel van Garrel (Warwick) - Open-closed duality in action

I will give an example-based talk describing open-closed duality and the log-local principle in enumerative geometry. Based on joint work in progress with Pierrick Bousseau and Andrea Brini.

Chris Lazda (Warwick) - The trace map for rigid analytic varieties and adic spaces

In my talk I will report on joint work in progress with Tomoyuki Abe on a formalism of compactly supported cohomology and duality for rigid analytic varieties. The key point is that by working instead with adic spaces, these varieties have canonical compactifications, which allows for a Grothendiec-style definition of compactly supported cohomology giving sensible answers even for affinoids such as the closed unit disc. Having such a well-behaved formalism enables the construction of a trace map for an arbitrary smooth morphism of (analytic) adic spaces, which then begs the question of to what extent Serre-Grothendieck duality holds for analytic varieties. If time permits I will also say a little bit about our motivation for studying these questions, which was to prove new comparison results in the theory of rigid cohomology.

Ivan Cheltsov (Edinburgh) - Toric G-solid Fano threefolds

Qingyuan Jiang (Edinburgh) - The geometry of (resolutions of) degeneracy loci

In this talk we will discuss the geometry, especially the derived categories and Chow groups (also Hodge structures), on certain canonical resolution spaces of degeneracy loci of maps between vector bundles. We will mainly focus on the classical examples of Brill-Noether theory on curves, and then explain the general results and the proofs through these examples.

Hyenho Lho (ETH Zurich) - Tautological relations on the moduli space of stable maps

I will review the brief history of tautological relations on the moduli space of curves. On the moduli space of smooth curves, Faber and Zagier conjectured set of relations among kappa classes which were proven by Pandharipande and Pixton. Later these relations were extended to the moduli space of stable curves by Pixton. These relations were proven by Pandharipande, Pixton and Zvonkine. I will explain how to describe these relations using stable graphs. I will sketch the idea of proof of Pixton’s relation due to Janda who used equivariant Gromov-Witten theory of projective line. After this I will explain how to get a set of tautological relations on the moduli space of stable maps to X which naturally generalise the Pixton’s relations. This talk is based on the joint work in progress with Younghan Bae.

Navid Nabijou (Glasgow) - Tangent curves to a smooth cubic, degenerations and blowups

It is well-known that every smooth plane cubic E supports precisely 9 flex lines; lines in P2 which intersect E in a single point. By analogy, we may ask: "How many degree d curves intersect E in a single point?" The problem of calculating the numbers of such tangent curves has fascinated enumerative geometers for decades. Despite being an extremely classical and concrete problem, it was not until the advent of Gromov-Witten theory in the 1990s that a general method was discovered.
Although we now have algorithms for computing these numbers, this is not the end of the story, since they also exhibit a huge amount of hidden structure. They have close connections to the enumerative theory of O(-E), via the local-logarithmic correspondence of van Garrel-Graber-Ruddat, and satisfy a remarkable recursive formula, conjectured by Takahashi and recently proven by Bousseau.
In this talk, I will discuss two distinct projects which take inspiration from this geometry. In joint work with Lawrence Barrott, we study the behaviour of the tangent curves as the cubic E degenerates to a cycle of lines. Using the machinery of logarithmic Gromov-Witten theory, we obtain detailed information concerning how these tangent curves degenerate along with the cubic. The resulting statements are purely classical, with no reference to Gromov-Witten theory, but they do not appear to admit a classical proof. In a separate project, joint with Dhruv Ranganathan, we use iterated blowups of moduli spaces to extend the local-logarithmic correspondence to the case of normal crossings divisors, bootstrapping from the smooth divisor case; this gives access to a large number of previously unknown logarithmic Gromov-Witten theories.

Clark Barwick (Edinburgh) - Compactifying arithmetic schemes

Half a century ago, Barry Mazur launched the industry known today as ‘arithmetic topology’ by observing that the ‘homotopy theory’ of a number ring O_F closely resembles that of a 3-manifold. Indeed, class field theory implies a kind of 3-dimensional Poincaré Duality for the étale cohomology of O_F. Unfortunately, this analogy has always suffered from deficiencies with real places and with 2-local coefficients. We propose a different, homotopical compactification, which enjoys a 3-dimensional duality theorem with no fudging around real places and the prime 2. The compactification is closely linked to the geometrisation of Galois and Weil groups, which we will also discuss.

Tiago Guerreiro (Loughborough) - On the birational non-rigidity of certain Fano 3-folds

The birational classification of algebraic varieties in dimension 3 saw a major development after the completion of the minimal model programme in the 1980s. Roughly speaking, given an algebraic variety, MMP produces a simpler model that is either a “Mori fibre space”, starting from a variety with negative Kodaira dimension, or a “minimal model” otherwise. The next natural step is to study relations among these outputs. In that spirit we show, mainly through examples, how to construct explicit birational maps between certain Fano 3-folds and other Mori fibre spaces.