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. For help joining a talk, please follow these instructions. 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). Recordings of talks are available.

Upcoming Talks

• • 6 August 2020, 10-11am
  • Yang-Hui He (City and Oxford)
  • Universes as big data: superstrings, Calabi-Yau manifolds and machine-learning
  • We review how historically the problem of string phenomenology lead theoretical physics first to algebraic/diffenretial geometry, and then to computational geometry, and now to data science and AI. With the concrete playground of the Calabi-Yau landscape, accumulated by the collaboration of physicists, mathematicians and computer scientists over the last 4 decades, we show how the latest techniques in machine-learning can help explore problems of physical and mathematical interest.
  • Link to Event
• • 13 August 2020, 4-5pm
  • Nathan Ilten (Simon Fraser)
  • Type D associahedra are unobstructed
  • Generalized associahedra associated to any root system were introduced by Fomin and Zelevinsky in their study of cluster algebras. For type A root systems, one recovers the classical associahedron parametrizing triangulations of a regular n-gon. For type D root systems, one obtains a polytope parametrizing centrally symmetric triangulations of a 2n-gon. In previous work, Jan Christophersen and I showed that the Stanley-Reisner ring of the simplicial complex dual to the boundary of the classical associahedron is unobstructed, that is, has vanishing second cotangent cohomology. This could be used to find toric degenerations of the Grassmannian G(2,n). In this talk, I will describe work-in-progress that generalizes this unobstructedness result to the type D associahedron.
  • Link to Event
• • 20 August 2020, 2-3pm
  • Man-Wai "Mandy" Cheung (Harvard)
  • Polytopes, wall crossings, and cluster varieties
  • Cluster varieties are log Calabi-Yau varieties which are a union of algebraic tori glued by birational "mutation" maps. Partial compactifications of the varieties, studied by Gross, Hacking, Keel, and Kontsevich, generalize the polytope construction of toric varieties. However, it is not clear from the definitions how to characterize the polytopes giving compactifications of cluster varieties. We will show how to describe the compactifications easily by broken line convexity. As an application, we will see the non-integral vertex in the Newton Okounkov body of Gr(3,6) comes from broken line convexity. Further, we will also see certain positive polytopes will give us hints about the Batyrev mirror in the cluster setting. The mutations of the polytopes will be related to the almost toric fibration from the symplectic point of view. Finally, we can see how to extend the idea of gluing of tori in Floer theory which then ended up with the Family Floer Mirror for the del Pezzo surfaces of degree 5 and 6. The talk will be based on a series of joint works with Bossinger, Lin, Magee, Najera-Chavez, and Vienna.
• • 27 August 2020, 1.30-2.30pm
  • Andrea Petracci (FU Berlin)
  • TBA
• • 3 September 2020, 3-4pm
  • Renato Vianna (Rio de Janeiro)
  • TBA
• • 4 September 2020, 3-4pm
  • Andrew Harder (Lehigh)
  • TBA
• • 10 September 2020, 4-5pm
  • Lara Bossinger (Oaxaca)
  • Families of Gröbner degenerations, Grassmannians, and universal cluster algebras
  • Let V be the weighted projective variety defined by a weighted homogeneous ideal J and C a maximal cone in the Gröbner fan of J with m rays. We construct a flat family over affine m-space that assembles the Gröbner degenerations of V associated with all faces of C. This is a multi-parameter generalization of the classical one-parameter Gröbner degeneration associated to a weight. We show that our family can be constructed from Kaveh-Manon's recent work on the classification of toric flat families over toric varieties: it is the pullback of a toric family defined by a Rees algebra with base X_C (the toric variety associated to C) along the universal torsor A^m -> X_C. If time permits I will explain how to apply this construction to the Grassmannians Gr(2,n) (with Plücker embedding) and Gr(3,6) (with "cluster embedding"). In each case there exists a unique maximal Gröbner cone whose associated initial ideal is the Stanley-Reisner ideal of the cluster complex. We show that the corresponding cluster algebra with universal coefficients arises as the algebra defining the flat family associated to this cone. Further, for Gr(2,n) we show how Escobar-Harada's mutation of Newton-Okounkov bodies can be recovered as tropicalized cluster mutation. This is joint work with Fatemeh Mohammadi and Alfredo Nájera Chávez.
• • 17 September 2020, 10-11am
  • Ronan Terpereau (Bourgogne)
  • Actions of connected algebraic groups on rational 3-dimensional Mori fibrations
  • In this talk we will study the connected algebraic groups acting on Mori fibrations X -> Y with X a rational threefold and Y a curve or a surface. We will see how these groups can be classified, using the minimal model program (MMP) and the Sarkisov program, and how our results make possible to recover most of the classification of the connected algebraic subgroups of the Cremona group Bir(P³) obtained by Hiroshi Umemura in the 1980's  when the base field is the field of complex numbers.

Past Talks

• • 30 July 2020, 3-4pm
  • Benjamin Braun (Kentucky)
  • The integer decomposition property and Ehrhart unimodality for weighted projective space simplices
  • We consider lattice simplices corresponding to weighted projective spaces where one of the weights is 1. We study the integer decomposition property and Ehrhart unimodality for such simplices by focusing on restrictions regarding the multiplicity of each weight. We introduce a necessary condition for when a simplex satisfies the integer decomposition property, and classify those simplices that satisfy it in the case where there are no more than three distinct weights. We also introduce the notion of reflexive stabilizations of a simpex of this type, and show that higher-order reflexive stabilizations fail to be Ehrhart unimodal and fail to have the integer decomposition property. This is joint work with Robert Davis, Morgan Lane, and Liam Solus.
  • Video, Slides
• • 24 July 2020, 4-5pm
  • Elana Kalashnikov (Harvard)
  • Constructing Laurent polynomial mirrors for quiver flag zero loci
  • All smooth Fano varieties of dimension at most three can be constructed as either toric complete intersections (subvarieties of toric varieties) or quiver flag zero loci (subvarieties of quiver flag varieties). Conjecturally, Fano varieties are expected to mirror certain Laurent polynomials. The construction of mirrors of Fano toric complete intersections is well-understood. In this talk, I'll discuss evidence for this conjecture by proposing a method of constructing mirrors for Fano quiver flag zero loci. A key step of the construction is via finding toric degenerations of the ambient quiver flag varieties. These degenerations generalise the Gelfand-Cetlin degeneration of flag varieties, which in the Grassmannian case has an important role in the cluster structure of its coordinate ring.
  • Video, Slides
• • 16 July 2020, 1.30-2.30pm
  • Hendrik Süß (Manchester)
  • Normalised volumes of singularities
  • The notion of the normalised volume of a singularity has been introduced relatively recently, but plays a crucial role in the context of Einstein metrics and K-stability. After introducing this invariant my plan is to specialise quickly to the case of toric singularities and show that even in this relatively simple setting interesting phenomena occur.
  • Video, Slides
• • 15 July 2020, 10-11am
  • Chris Lazda (Warwick)
  • A Neron-Ogg-Shafarevich criterion for K3 surfaces
  • The naive analogue of the Néron-Ogg-Shafarevich criterion fails for K3 surfaces, that is, there exist K3 surfaces over Henselian, discretely valued fields K, with unramified etale cohomology groups, but which do not admit good reduction over K. Assuming potential semi-stable reduction, I will show how to correct this by proving that a K3 surface has good reduction if and only if is second cohomology is unramified, and the associated Galois representation over the residue field coincides with the second cohomology of a certain "canonical reduction" of X. This is joint work with B. Chiarellotto and C. Liedtke.
  • Video, Slides
• • 8 July 2020, 1.30-2.30pm
  • Ed Segal (UCL)
  • Semi-orthogonal decompositions and discriminants
  • The derived category of a toric variety can usually be decomposed into smaller pieces, by passing through different birational models and applying the "windows" theory relating VGIT and derived categories. There are many choices involved and the decompositions are not unique. We prove a Jordan-Holder result, that the multiplicities of the pieces are independent of choices. If the toric variety is Calabi-Yau then there are no decompositions, instead the theory produces symmetries of the derived category. Physics predicts that these all these symmetries form an action of the fundamental group of the "FI parameter space". I'll explain why our Jordan-Holder result is necessary for this prediction to work, and state a conjecture (based on earlier work of Aspinwall-Plesser-Wang) relating our multiplicities to the geometry of the FI parameter space. This is joint work with Alex Kite.
  • Video, Slides
• • 2 July 2020, 1.30-2.30pm
  • Gregory Smith (Queen’s)
  • Geometry of smooth Hilbert schemes
  • How can we understand the subvarieties of a fixed projective space? Hilbert schemes provide the geometric answer to this question. After surveying some features of these natural parameter spaces, we will classify the smooth Hilbert schemes. Time permitting, we will also describe the geometry of nonsingular Hilbert schemes by interpreting them as suitable generalisations of partial flag varieties. This talk is based on joint work with Roy Skjelnes (KTH).
  • Video, Slides
• • 25 June 2020, 10-11am
  • Klaus Altmann (FU Berlin)
  • Displaying the cohomology of toric line bundles
  • Line bundles L on projective toric varieties can be understood as formal differences (Δ⁺-Δ^-) of convex polyhedra in the character lattice. We show how it is possible to use this language for understanding the cohomology of L by studying the set-theoretic difference (Δ^-\Δ⁺). Moreover, when interpreting these cohomology groups as certain Ext-groups, we demonstrate how the approach via (Δ^-\Δ⁺) leads to a direct description of the associated extensions. The first part is joint work with Jarek Buczinski, Lars Kastner, David Ploog, and Anna-Lena Winz; the second is work in progress with Amelie Flatt.
  • Video
• • 18 June 2020, 1.30-2.30pm
  • Leonid Monin (Bristol)
  • Inversion of matrices, a C* action on Grassmannians and the space of complete quadrics
  • Let Γ be the closure of the set of pairs (A,A^-1) of symmetric matrices of size n. In other words, Γ is the graph of the inversion map on the space Sym_n of symmetric matrices of size n. What is the cohomology class of Γ in the product of projective spaces? Equivalently, what is the degree of the variety L^-1 obtained as the closure of the set of inverses of matrices from a generic linear subspace L of Sym_n? This question is interesting in its own right but it is also motivated by algebraic statistics. In my talk, I will explain how to invert a matrix using a C* action on Grassmannians, relate the above question to classical enumerative problems about quadrics, and give several possible answers. This is joint work in progress with Laurent Manivel, Mateusz Michalek, Tim Seynnaeve, Martin Vodicka, Andrzej Weber, and Jaroslaw A. Wisniewski.
  • Video
• • 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.
  • Video, Slides
• • 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.
  • Video
• • 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.
  • Video, Slides
• • 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.
  • Video, Slides
• • 19 May 2020, 2-3pm
  • Timothy 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.
  • Video
• • 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.
  • Video
• • 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.
  • Video, Slides
• • 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.
  • Slides
• • 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.
  • Slides

Past Reading Groups

• • 31 March - 7 July 2020, 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.