Online Algebraic Geometry Seminar

This seminar is held online via Microsoft Teams on the Nottingham University "Algebraic Geometry" team. To be added to this team, please email one of Alex Kasprzyk, Livia Campo, or Johannes Hofscheier. Everybody is welcome.

Note: All times are UK times.

Subscribe to the online calendar to be kept up-to-date about upcoming talks (also available in iCal format).

Reading Group

• • Tuesdays, 9.30-10.30am
  • Geometry of Numbers
  • A weekly reading group working through Siegel's "Lectures on the Geometry of Numbers". Material presented by Livia Campo, Daniel Cavey, Giulia Codenotti, Oliver Daisey, Thomas Hall, Johannes Hofscheier, Katharina Jochemko, and Leonid Monin.

Upcoming Talks

• • 28 May 2020, 10-11am
  • Tom Sutherland (Lisbon)
  • Mirror symmetry for Painlevé surfaces
  • This talk will survey aspects of mirror symmetry for ten families of non-compact hyperkähler manifolds on which the dynamics of one of the Painlevé equations is naturally defined. They each have a pair of natural realisations: one as the complement of a singular fibre of a rational elliptic surface and another as the complement of a triangle of lines in a (singular) cubic surface. The two realisations relate closely to a space of stability conditions and a cluster variety of a quiver respectively, providing a perspective on SYZ mirror symmetry for these manifolds.
• • 4 June 2020, 1.30-2.30pm
  • Giuliano Gagliardi (Hannover and MPI Bonn)
  • The Manin-Peyre conjecture for smooth spherical Fano varieties of semisimple rank one
  • The Manin-Peyre conjecture is established for a class of smooth spherical Fano varieties of semisimple rank one. This includes all smooth spherical Fano threefolds of type T as well as some higher-dimensional smooth spherical Fano varieties. This is joint work with Valentin Blomer, Jörg Brüdern, and Ulrich Derenthal.
• • 11 June 2020, 10-11am
  • Karin Schaller (FU Berlin)
  • Polyhedral Divisors and Orbit Decompositions of Normal Affine Varieties with Torus Action
  • Normal affine varieties of dimension n with an effective action of a k-dimensional algebraic torus can be described completely in terms of proper polyhedral divisors living on semiprojective varieties of dimension n−k. We use the language of polyhedral divisors to study the collection of T-orbits and T-orbit closures of a normal affine T-variety in terms of its defining pp-divisor. This is based on previous work of Klaus Altmann and Jürgen Hausen complemented by work in progress with Klaus Altmann.
• • 18 June 2020, 1.30-2.30pm
  • Leonid Monin (Bristol)
  • TBA
• • 25 June 2020, 10-11am
  • Klaus Altmann (FU Berlin)
  • TBA

Past Talks

• • 21 May 2020, 1.30-2.30pm
  • Jesús Martinez Garcia (Essex)
  • The moduli continuity method for log Fano pairs
  • The moduli continuity method, pioneered by Odaka, Spotti and Sun, allows us to explicitly provide algebraic charts of the Gromov-Hausdorff compactification of (possibly singular) Kähler-Einstein metrics. Assuming we can provide a homeomorphism to some 'known' algebraic compactification (customarily, a GIT one) the method allows us to determine which Fano varieties (or more generally log Fano pairs) are K-polystable in a given deformation family. In this talk we provide the first examples of compactification of the moduli of log Fano pairs for the simplest deformation family: that of projective space and a hypersurface, and mention related results for cubic surfaces. This is joint work with Patricio Gallardo and Cristiano Spotti.
• • 19 May 2020, 2-3pm
  • Tim Logvinenko (Cardiff)
  • Skein-triangulated representations of generalised braids
  • Ordinary braid group Br_n is a well-known algebraic structure which encodes configurations of n non-touching strands ("braids") up to continuous transformations ("isotopies"). A classical result of Khovanov and Thomas states that there is a natural categorical action of Br_n on the derived category of the cotangent bundle of the variety of complete flags in Cⁿ. In this talk, I will introduce a new structure: the category GBr_n of generalised braids. These are the braids whose strands are allowed to touch in a certain way. They have multiple endpoint configurations and can be non-invertible, thus forming a category rather than a group. In the context of triangulated categories, it is natural to impose certain relations which result in the notion of a skein-triangulated representation of GBr_n. A decade-old conjecture states that there a skein-triangulated action of GBr_n on the cotangent bundles of the varieties of full and partial flags in Cⁿ. We prove this conjecture for n = 3. We also show that any categorical action of Br_n can be lifted to a skein-triangulated action of GBr_n, which behaves like a categorical nil Hecke algebra. This is a joint work with Rina Anno and Lorenzo De Biase.
• • 14 May 2020, 10-11am
  • Alan Thompson (Loughborough)
  • Threefolds fibred by sextic double planes
  • I will discuss the theory of threefolds fibred by K3 surfaces mirror to the sextic double plane. This theory is unexpectedly rich, in part due to the presence of a polarisation-preserving involution on such K3 surfaces. I will present an explicit birational classification result for such threefolds, along with computations of several of their basic invariants. Along the way we will uncover several (perhaps) surprising links between this theory and Kodaira's theory of elliptic surfaces. This is joint work with Remkes Kooistra.
• • 13 May 2020, 1-2pm
  • Tom Ducat (Imperial)
  • A Laurent phenomenon for OGr(5,10) and explicit mirror symmetry for the Fano 3-fold V₁₂
  • 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 dP₅. I will briefly recap 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 a completely explicit K3 fibration which is mirror to the Fano 3-fold V₁₂, as well as some other Fano 3-folds.
• • 6 May 2020, 10-11am
  • Livia Campo (Nottingham)
  • On a high pliability quintic hypersurface
  • In this talk we exhibit an example of a quintic hypersurface with a certain compound singularity that has pliability at least 2. We also show that, while a non-trivial sequence of birational transformations can be constructed between the two elements of the pliability set, the Sarkisov link connecting them is not evident. This is done by studying birational links for codimension 4 index 1 Fano 3-folds having Picard rank 2.
• • 30 April 2020, 9.30-10.30am
  • Florian Kohl (Aalto)
  • Unconditional reflexive polytopes
  • A convex body is unconditional if it is symmetric with respect to reflections in all coordinate hyperplanes. In this talk, we investigate unconditional lattice polytopes with respect to geometric, combinatorial, and algebraic properties. In particular, we characterise unconditional reflexive polytopes in terms of perfect graphs. As a prime example, we study a type-B analogue of the Birkhoff polytope. This talk is based on joint work with McCabe Olsen and Raman Sanyal.
• • 16 April 2020, 10-11am
  • Alessio Corti (Imperial)
  • Volume preserving birational selfmaps of P³
  • I describe some results on the structure of the group Bir(P³, D) where D is a quartic surface with mild singularities. Work with Carolina Araujo and Alex Massarenti.
• • 8 April 2020, 10-11am
  • Thomas Prince (Oxford)
  • Perturbing torus fibrations on threefold singularities
  • Fix a Q-Gorenstein threefold toric singularity X determined by a rational polyhedral cone C in N_R, together with a collection of piecewise linear maps (or combinatorial mutations) T₁,...,T_k such that the image of C* under the composition of these linear maps is a half space in R³. We describe how to perturb the torus fibration X to C*, whose fibres are orbits of the (toric) T³ action on X, to a torus fibration on a space X' which is a manifold away from a finite collection of singular points. Around each of these singular points X' has the structure of a terminal cyclic quotient singularity. We outline how to globalise this to construct torus fibrations over 3-dimensional balls which correspond to (partially) smoothing a three-dimensional toric Fano variety to a Fano variety with cyclic quotient terminal singularities. The combinatorial input to this process is closely related to the notion of zero-mutable Laurent polynomial which has been recently studied by Corti, Kasprzyk, and Pitton.
• • 2 April 2020, 10-11am
  • Alice Cuzzucoli (Warwick)
  • A glimpse at the classification of orbifold del Pezzo surfaces
  • In this talk, we will discuss the main ingredients involved in the classification of del Pezzo surfaces with orbifold points, i.e. complex projective varieties of dimension two admitting log terminal singularities. In the smooth case, we have a well-known birational classification dating back to the 19th century. In the singular case, we are still missing a classification just as complete. Nevertheless, in the case of cyclic quotient singularities, we have some interesting constructions. We will introduce the most crucial aspects of such constructions, which are divided into three main steps: firstly, by analysing the graded rings of such surfaces, we can find a bound on the number of singularities and the relative invariants; secondly, with the help of Mori Theory, we can give a first representation of our birational models; then, by having a brief look at the toric case, we will describe how toric degenerations come into play in this classification. Ultimately, we can recreate analogous constructions to the cascade of blow ups for the smooth case with the representatives of specific deformation classes of our orbifolds.