East Midlands Seminar in Geometry
The East Midlands Seminar in Geometry (EmSG) is an Algebraic Geometry seminar based at the Universities of Nottingham, Loughborough, Leicester, and Sheffield. The EmSG meets approximately six times per year, and is funded by a grant from the LMS.
The general organisers are Alexander Kasprzyk (Nottingham), Artie Prendergast-Smith (Loughborough), Frank Neumann (Leicester), and Paul Johnson (Sheffield).
Subscribe to the online calendar:
- 3 March 2017
- 12-1pm: Gavin Brown (Warwick); Michael Atiyah Building MA119 (First Floor)
- Fanos in P2xP2 format
- The classification of complex (projective) curves and surfaces is ancient (1850s, 1905, 1960s). In 3 dimensions we can say an enormous amount, but nothing like an explicit classification yet. The general shape of classification is familiar: just as with compact real orientable surfaces there are three broad classes: Fano 3-folds, Calabi-Yau 3-folds and 3-folds of general type (the analogues of positively curved spheres, flat tori and hyperbolic g-holed tori for g at least 2). Fano 3-folds can be embedded in weighted projective space (which is the quotient of usual projective space by a finite abelian group) in their total anticanonical embedding. We know the Hilbert series of all possible such embeddings (including, sadly, many that surely don’t exist). In low codimension (<= 3 or 4ish) those involved in Miles Reid’s graded rings program have found a few hundred deformation families of Fano 3-folds to realise all the Hilbert series in those cases. Sometimes more than one deformation family may realise a given Hilbert series. I will describe some families of Fano 3-folds whose equations look like those of the Segre embedding of P2xP2 (so lie in codimension 4) that seem to be new. I’d like to make this all very explicit: these varieties are the loci where general 3x3 matrices drop rank, and much of the birational geometry I would like to explain can be described in terms of hands-on linear algebra. (This is in progress, and joint with Al Kasprzyk and Imran Qureshi.)
- 2-3pm: Liana Heuberger (Imperial); Attenborough Seminar Block LR 001 (Ground Floor)
- Del Pezzo surfaces with 1/3(1,1) singularities
- I will talk about how and why we classify these surfaces, resulting in 29 Q-Gorenstein deformation families. We will discuss their biregular invariants and give a gentle introduction to the Minimal Model Program needed for the classification. To go more in depth with their description, we look at their model constructions (mostly) as complete intersections in toric varieties. This is joint work with Alessio Corti.
- 3.30-4.30pm: Benjamin Nill (Magdeburg); Attenborough Seminar Block LR 001 (Ground Floor)
- On the maximal degree of toric Fano varieties with canonical singularities
- Toric Fano varieties are among the most studied classes of toric varieties. This is due to their explicit description in terms of certain polytopes, called Fano polytopes, and the importance of Fano varieties in algebraic geometry. Here, a Fano polytope is a convex polytope with the origin in its interior where each vertex has coprime integer coordinates. By the toric dictionary, interesting invariants of toric Fano varieties (such as the anticanonical degree) have often a combinatorial description (such as the volume). In this talk, I will present joint work with Gabriele Balletti and Alexander Kasprzyk where we give a sharp upper bound on the anticanonical degree of toric Fano varieties with canonical singularities. I will describe the geometry of numbers in the proof, some related results, and some challenging questions that are still wide open.
- 18 January 2017
- Sheffield, Hicks Building, F24
- 1-2pm: Anna Felikson (Durham)
- Geometric realisations of quiver mutations
- A quiver is a weighted oriented graph, a mutation of a quiver is a simple combinatorial transformation arising in the theory of cluster algebras. In this talk we connect mutations of quivers to reflection groups acting on linear spaces and to groups generated by point symmetries in the hyperbolic plane. We show that any mutation class of rank 3 quivers admits a geometric presentation via such a group and that the properties of this presentation are controlled by the Markov constant p2+q2+r2-pqr, where p,q,r are the weights of the arrows in the quiver. This is a joint work with Pavel Tumarkin.
- 2.30-3.30pm: Ed Segal (Imperial)
- Homological projective duality for Pfaffians
- A Pfaffian variety is a space of anti-symmetric matrices with some fixed upper bound on the rank, and the projective dual of a Pfaffian variety is a another Pfaffian variety. Kuznetsov conjectured that these varieties should have non-commutative resolutions, with the derived categories of projectively-dual pairs satisfying a nice relationship called "homological projective duality". I will explain what all of the above means, and describe the construction of these non-commutative resolutions (which is due to Spenko and Van den Bergh) and a proof that they do indeed satisfy the duality. Our proof is motivated by a physical duality of non-abelian GLSMs, proposed by Hori. This is joint work with Jorgen Rennemo.
- 4-5pm: Clelia Pech (Kent)
- Rational curves on some varieties with an action of an algebraic group
- In this talk I will report on joint work in progress with R. Gonzales, N. Perrin and A. Samokhin on rational curves on a family of two-orbit varieties. One of the motivations is to study the quantum cohomology of these spaces, an associative and commutative deformation of their usual cohomology ring whose structure constants are given by counts of rational curves. Using a classification of these varieties by B. Pasquier we study their moduli spaces of rational curves and deduce a Chevalley-type formula for the quantum cup product by the hyperplane class. Some of the varieties we consider have particularly well-behaved moduli spaces of stable maps, and in these cases we obtain a more precise description of the quantum cohomology.
- 15 November 2016
- Warwick, Mathematics Institute
- 12-1pm: Joe Karmazyn (Bath); Room B3.01
- Simultaneous resolutions and noncommutative algebras
- Minimal resolutions of surface quotient singularities can be studied and understood via noncommutative algebra in a variety of ways packaged as the McKay correspondence. Assorted higher dimensional examples, such as flopping contractions of 3-folds, can be realised from simultaneous resolutions associated to surface quotient singularities. I will discuss how these simultaneous resolutions can also be understood via noncommutative algebras, and how certain examples can be easily calculated.
- 2-3pm: Elisa Postinghel (Loughborough); Room MS.04
- Tropical compactifications, Mori dream spaces and Minkowski bases
- Joint work in progress with Stefano Urbinati. Given a Mori Dream Space X, we construct via tropicalisation a model dominating all the small QQ-factorial modifications of X. Via this construction we recover a Minkowski basis for the Newton-Okounkov bodies of Cartier divisors on X and hence generators of the movable cone of X.
- 4-5pm: Robert Marsh (Leeds); Room B3.02
- Twists of Plücker coordinates as Dimer partition functions
- Joint work with Jeanne Scott. The homogeneous coordinate ring of the Grassmannian Gr(k,n) has a cluster algebra structure defined in terms of certain planar diagrams known as Postnikov diagrams. The cluster associated to such a diagram consists entirely of Plücker coordinates. We introduce a twist map on Gr(k,n), related to the twist of Berenstein-Fomin-Zelevinsky, and give an explicit Laurent expansion for the twist of an arbitrary Plücker coordinate in terms of a scaled matching polynomial. This matching polynomial arises from the bipartite graph dual to the Postnikov diagram of the initial cluster, modified by an appropriate boundary condition.
- 5.15-6.15pm: Thomas Prince (Imperial); Room B3.02
- A symplectic approach to polytope mutation
- Polytope mutation is a combinatorial operation which appeared in the study of the birational geometry of Landau-Ginzburg models mirror-dual to Fano manifolds. We give a mirror/symplectic account of this subject. This heavily utilizes the notion of an almost-toric Lagrangian fibration, due to Symington. This perspective elucidates the connection with cluster and quiver mutation (in the surface case) as well as the connection to toric degenerations via "algebraization" techniques due to Gross-Siebert.
- 9 November 2016
- Nottingham, Physics C5
- 1-2pm: Elana Kalashnikov (Imperial)
- Some new Fano fourfolds
- I will discuss some new Fano fourfolds found in quiver flag varieties as zero loci of sections of homogenous vector bundles. This is work with Tom Coates. The classification of Fano varieties up to deformation is known in dimensions 1, 2, and 3. The program of Coates, Corti, Galkin, Golyshev, Kasprzyk, and others seeks to use mirror symmetry to find and classify Fano varieties in dimension 4 (and more). Küchle classified 4 dimensional Fano varieties of this form in Grassmannians; quiver flag varieties are a generalisation of Grassmannians that include flag varieties as well. I will briefly discuss mirror symmetry for Fano varieties, then explore some of the combinatorial aspects of quiver flag varieties which make them a good testing ground for the conjectures and the search, and finally describe the 138 new Fano fourfolds we have found so far.
- 2.15-2.45pm: Ivan Cheltsov (Edinburgh)
- Del Pezzo surfaces, ample line bundles on them and their alpha-invariants
- This "pre-talk" is aimed at PhD students and non-experts.
- 3-4pm: Ivan Cheltsov (Edinburgh)
- Stable and unstable polarized del Pezzo surfaces
- Yau-Tian-Donaldson conjecture, recently proved by Chen, Donaldson and Sun, says that a Fano manifold is Kahler-Einstein if and only if it is K-stable. Its stronger form, still open, says that a polarized manifold (M,L) is K-stable if and only if M admits a constant scalar curvature with Kahler class in L. In my talks, I will describe K-stability of ample line bundles on smooth del Pezzo surfaces (two-dimensional Fano manifolds). I will show how to apply recent result of Dervan to prove K-stability and how to use flop-version of Ross and Thomas's obstruction to prove instability. This is joint work with Jesus Martinez-Garcia.
- 4.15-5.15pm: Jonny Evans (UCL)
- Lagrangian cell complexes and Markov numbers
- Joint work with Ivan Smith. In a degenerating family of complex projective varieties, with one singular member, there are cohomology classes ("vanishing cycles") in the smooth fibre which disappear in the singular fibre. These vanishing cycles are more than just cohomology classes: they're realised geometrically by Lagrangian subsets of the degenerating smooth variety. For example, the vanishing cycle of a nodal degeneration is a Lagrangian sphere. In this talk I will focus on the vanishing cycles of a class of surface singularities called Wahl singularities. Their vanishing cycles are cell complexes (we call them "pinwheels"). We deduce constraints on configurations of Wahl singularities in degenerations of CP^2 from nondisplaceability properties of the Lagrangian vanishing cycles. In particular, we give a symplectic topology explanation for the appearance of Markov numbers in this problem.
- 2 November 2016
- Loughborough, Schofield Building, Room 1.01
- 1-2pm: Alan Thompson (Warwick)
- Constructing threefolds via K3 fibrations
- Kodaira's and Tate's classification of elliptic surfaces is a powerful result with many applications. In brief, it states that an elliptic surface is essentially determined by two invariants: the functional invariant, which controls the behaviour away from a few special points, and the homological invariant, which determines the fibres appearing over the special points. An analogous result for threefolds fibred by K3 surfaces would be quite desirable but is, at least with current technology, completely intractable. However, if one simplifies the problem by restricting to certain classes of lattice polarised K3 surfaces, then there is much more that one can say; in particular, there are analogues of the functional and homological invariants, which play similar roles to their surface counterparts. I will present a series of results pertaining to the construction of threefolds fibred by lattice polarised K3 surfaces, and show how this may be used to study Calabi-Yau threefolds admitting K3 fibrations. This talk is on joint work with Charles Doran, Andrew Harder, and Andrey Novoseltsev.
- 2.30-3.30pm: Diletta Martinelli (Edinburgh)
- On the number of minimal models of a smooth threefold of general type
- Finding minimal models is the first step in the birational classification of smooth projective varieties. After it is established that a minimal model exists some natural questions arise such as: is it the minimal model unique? If not, how many are they? After recalling all the necessary notions of the Minimal Model Program, I will explain that varieties of general type admit a finite number of minimal models. I will talk about a recent joint project with Stefan Schreieder and Luca Tasin where we prove that in the case of threefolds this number is bounded by a constant depending only on the Betti numbers. I will also show that in some cases it is possible to compute this constant explicitly.
- 4-5pm: Miles Reid (Warwick)
- The Tate-Oort group scheme of order p
- The aim is mostly group schemes in mixed characteristic, but the methods are mostly Galois theory of cyclotomic fields.
- 12-14 September 2016
- Nottingham, Physics C5
- The different faces of geometry, a workshop in honour of Fedor Bogomolov
A workshop dedicated to Fedor Bogomolov on the occasion of his 70th birthday. Speakers are:
- Ekaterina Amerik (HSE, Moscow)
- Christian Böhning (Univ. Warwick)
- Paolo Cascini (ICL)
- Ivan Cheltsov(Univ. Edinburgh)
- Ivan Fesenko(Univ. Nottingham)
- Mikhail Kapranov (IPMU, Tokyo)
- Ludmil Katzarkov (Univ. Wien)
- Kobi Kremnitzer (Univ. Oxford)
- Sergey Oblezin (Univ. Nottingham)
- Tony Pantev (Univ. Pennsylvania)
- Yuri Tschinkel (Courant Inst.)
- Misha Verbitsky (HSE, Moscow)
- Boris Zilber (Univ. Oxford)
- 13 May 2016
- Leicester, Michael Atiyah Building, First Floor, Room 119
- 12.30-1.30pm: Katrin Leschke (Leicester)
- Quaternionic Holomorphic Geometry
- In my talk, I will give a short introduction to Quaternionic Holomorphic Geometry: conformal maps into 3-space can be used used as an analogue for complex holomorphic functions. As an example of the theory I will discuss the Darboux transformation of minimal surfaces.
- 2.30-3.30pm: Marina Logares (Oxford)
- Higgs bundles, Integrable systems, singularities and a Torelli theorem
- We will introduce Higgs bundles over a punctured Riemann surface. The moduli space of such objects describes an integrable system that completely determines the Riemann surface together with the punctures. Hence, it provides a Torelli type theorem for such moduli spaces. This is joint work with I. Biswas and T. Gómez.
- 4-5pm: Gabriele Balletti (Stockholm)
- Classifications of lattice polytopes and open questions in Ehrhart Theory
- A lattice polytope P is the convex hull of finitely many points of a lattice (such as Z^d). Counting the lattice points of P leads to a discrete version of the volume of P. The Ehrhart Theory studies the relation between the discrete and the usual notion of volume of a polytope, but the most important problems in this area are still open already in dimension 3. In this talk I give an introduction to this theory and explain how classifications of lattice polytopes can suggest some behaviours in higher dimension.
- 18 March 2016
- Sheffield, Hicks Building J11
- 2-3pm: Diane Maclagan (Warwick)
- Tropical ideals, varieties, and schemes
- 3.30-4.30pm: Hendrik Suess (Manchester)
- Torus equivariant K-stability in complexity one
- 5 February 2016
- Loughborough, Schofield Building
- 1.30-2.30pm: Norbert Pintye (Loughborough)
- Complexity in Light of the Castelnuovo--Mumford Regularity
- 3-4pm: Roberto Svaldi (Cambridge)
- Hyperbolicity for log pairs
- A classical result in birational geometry, Mori's Cone Theorem, implies that if the canonical bundle of a variety X is not nef then X contains rational curves. This is the starting point of the so-called Minimal Model Program. In particular, hyperbolic varieties are positive from the point of view of birational geometry. Very much in the same vein, one could ask what happens for a quasi projective variety, Y. Using resolution of singularity, then one is lead to consider pairs (X, D) of a variety and a divisor, such that Y=X \ D. I will show how to obtain a theorem analogous to Mori's Cone Theorem in this context. Instead of rational complete curves, algebraic copies of the complex plane will make their appearance. I will also discuss an ampleness criterion for hyperbolic pairs.
- 16 December 2015
- Nottingham, Physics C04
- 2-3pm: Paul Johnson (Sheffield)
- Topology of Hilbert schemes and the Combinatorics of Partitions
- The Hilbert scheme of n points on a complex surface is a smooth manifold of dimension 2n. Their topology has beautiful structure related to physics, representation theory, and combinatorics. For instance, Göttsche's formula gives a product formula for generating functions for their Betti numbers. Hilbert schemes of points on C^2/G, for G a finite group, are also smooth, and when G is abelian their topology is encoded in the combinatorics of partitions. When G is a subgroup of SL_2, the topology is well understood and in terms of cores and quotients of partitions. Following Gusein-Zade, Luengo and Melle-Hernández we study general abelian G, stating a conjectural product formula, and proving a homological stability result using a generalization of cores and quotients.
- 3.30-4.30pm: Cristina Manolache (Imperial)
- Enumerative meaning of genus one Gromov-Witten invariants
- Enumerative questions have a very long history in Mathematics and have been subject to a significant revival in the nineties with the construction of the moduli space of stable maps and the machinery allowing us to integrate on these very singular spaces. However, moduli spaces of stable maps have many "unwanted" components which are reflected in the intersection numbers. In this talk I will discuss the meaning of genus one Gromov-Witten invariants of three folds.
- 5-6pm: Lino Amorim (Oxford)
- Derived Lagrangian correspondences
- I will describe the construction of the Weinstein symplectic category of Lagrangian correspondences in the context of shifted symplectic geometry. I will explain how this follows from "quantizing" (-1)-shifted symplectic derived stacks: we assign a perverse sheaf to each (-1)-shifted symplectic derived stack (already done by Joyce and his collaborators) and a map of perverse sheaves to each (-1)-shifted Lagrangian correspondence (still conjectural).
Claiming back travel expenses
Travel expenses are covered by a grant from the LMS. Download and complete an expenses form, then post the completed form (along with receipts) to:
Getting to Nottingham
- Alexander Kasprzyk
- School of Mathematical Sciences
- University of Nottingham
- University Park
- NG7 2RD
There are trams leaving from Nottingham train station (go upstairs) every 7 minutes. You want a tram in the direction of Toton Lane; get off at the University of Nottingham stop. You need to buy a ticket from the machine before you get on the tram. The trip takes about 15 minutes. People travelling from Loughborough might prefer to catch the train to Beeston train station and walk to the campus.
The Maths building is number 20 on the campus map. Talks are usually held in the Physics building, opposite the Maths building, number 22 on the map. The tram stop is the green circle on the map near the South Entrance. A nice place for lunch is the Lakeside Arts cafe (number 49 on the map, near the lake and tram stop).
Getting to Loughborough
Talks are typically held in the Schofield Building.
From the station, catch the Kinchbus Sprint to the university (every 10 minutes; £1.90 each way). Get off at "Computer Studies"; the Schofield Building is directly behind you. Alternatively, the university is 30-40 minutes from the station on foot.
Getting to Leicester
Talks are typically held in the Michael Atiyah Building.
Getting to Sheffield
Talks are held in the Hicks Building.