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 Scheme 3 grant from the LMS.
The general organisers are Alexander Kasprzyk (Nottingham), Artie PrendergastSmith (Loughborough), Frank Neumann (Leicester), and Paul Johnson (Sheffield).
Next event
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 14 December 2017
 Leicester, Bennett Building Lecture Theatre 5 (BEN LT5)

 11am12pm: Anna Barbieri (Sheffield)
 Geometric structures on some spaces of stability conditions
 The space Stab(Q) of (Bridgeland) stability conditions on a category associated with a quiver Q is a complex manifold, whose geometry is partly governed by the combinatorics of the quiver, and there are welldefined invariants counting semistable objects. In a joint work with J. Stoppa and T. Sutherland, we show that, in some nice cases, these data endow Stab(Q) with the structures of Frobenius and Frobeniuslike manifold. I will discuss the main ingredients of this picture and the result for the quiver A_n.
 23pm: Hipolito Treffinger (Leicester)
 Giving an algebraic description of the support of scattering diagrams
 Scattering diagrams were introduced by Kontsevich, Soibelman and Gross, Hacking, Keel and Kontsevich in order to tackle the phenomenon of mirror symmetry, resulting, at the same time, in proofs for several conjectures in the theory of cluster algebras. Later, Bridgeland defined scattering diagrams for quivers with potential and he showed that both notions of scattering diagrams coincide under some technical condition on the potential. In this talk, following Bridgeland's methods, we study the wall and chamber structure of an algebra using the \tautilting theory introduced by Adachi, Iyama and Reiten. In particular we show how \taurigid pairs induce stability conditions \theta, as defined by King, and we give an explicit description of the category of \thetasemistable modules. As a corollary we show how \tautilting pairs induce chambers in the wall and chamber structure of the algebra. This includes all reachable chambers. Time permitting, we will talk about cvectors for every finite dimensional algebra and their categorical meaning.
 3.304.30pm: Jon Woolf (Liverpool)
 Stability conditions: phases and masses
 Each triangulated category T has an associated complex manifold Stab(T) of stability conditions. In general Stab(T) is difficult to compute, and only a few examples are understood in detail. I will discuss how properties of the phase distribution of a stability condition in Stab(T) relate to the complexity of the category T. In the simplest case, roughly when all stability conditions in Stab(T) have discrete phase distributions, this can be used to show that Stab(T) is contractible. This is joint work with Yu Qiu. I will also outline an approach to partially compactifying Stab(T) by allowing the masses of objects to vanish. Conjecturally, one obtains a space stratified by components of Stab(T/N) for various thick subcategories N of T. This is joint work with Nathan Broomhead, David Pauksztello, and David Ploog.
 4.305.30pm: David Pauksztello (Lancaster)
 Silting theory and stability spaces
 In this talk I will introduce the notion of silting objects and mutation of silting objects. I will then show how the combinatorics of silting mutation can give one information regarding the structure of the space of stability conditions. In particular, I will show how a certain discreteness of this mutation theory enables one to employ techniques of Qiu and Woolf to obtain the contractibility of the space of stability conditions for a class of mainstream algebraic examples, the socalled siltingdiscrete algebras. This talk will be a discussion of joint work with Nathan Broomhead, David Ploog, Manuel Saorin and Alexandra Zvonareva.
Past events

 1 November 2017
 Nottingham, Physics C27

 1.302.30pm: Karin Schaller (Tübingen)
 Stringy Chern classes of toric varieties, lattice polytopes, and the number 24
 We give a combinatorial interpretation of the stringy LibgoberWood identity in terms of generalised stringy Hodge numbers and intersection products of stringy Chern classes for arbitrary projective QGorenstein toric varieties. As a first application we derive a novel combinatorial identity relating reflexive polytopes of dimension d >= 4 to the number 24. Our second application is motivated by computations of stringy invariants of nondegenerated hypersurfaces in 3dimensional algebraic tori whose minimal models are K3 surfaces, giving rise to a combinatorial identity for the Euler number 24. Using combinatorial interpretations of the stringy Efunction and the stringy LibgoberWood identity, we show with purely combinatorial methods that this identity holds for any 3dimensional lattice polytope containing exactly one interior lattice point.
 2.453.45pm: Elena Gal (Oxford)
 Higher Hall algebras
 We recall the notion of a Hall algebra associated to a category, and explain how this construction can be done in a way that naturally includes a higher algebra structure, motivated by work of Toen and DyckerhoffKapranov. We will then explain how this leads to new insights about the bialgebra structure and related concepts.
 45pm: Rosemary Taylor (Warwick)
 Constructions of Fano 3folds using unprojections
 The Graded Ring Database uses numerical data to create a list of the Fano 3folds in weighted projective space which could exist. In codimensions 1, 2 and 3 we know those that exist. But what do we know of codimension 4? Unprojections provide a method for constructing explicit examples of these Fano varieties. This talk will provide an overview of the current research, beginning with an introduction to unprojections and concluding with recent progress in type II1 unprojections.

 18 October 2017
 Loughborough, Schofield 1.01

 1.302.30pm: Dylan Allegretti (Sheffield)
 Meromorphic projective structures and framed local systems
 A projective structure on an oriented surface S is an atlas of charts mapping open subsets of S into the Riemann sphere. There is a natural map from the space of projective structures to the PGL(2,C) character variety of S which sends a projective structure to its monodromy representation. In this talk, I will describe a meromorphic analog of this construction. I will introduce a moduli space parametrising projective structures with poles at a discrete set of points. I will explain how, in this setting, the object parametrising monodromy data is a type of cluster variety.
 2.453.45pm: Emilie Dufresne (Nottingham)
 Separating invariants and local cohomology
 The study of separating invariants is a new trend in Invariant Theory and a return to its roots: invariants as a classification tool. For a finite group acting linearly on a vector space, a separating set is simply a set of invariants whose elements separate the orbits o the action. Such a set need not generate the ring of invariants. In this talk, we give lower bounds on the size of separating sets based on the geometry of the action. These results are obtained via the study of the local cohomology with support at an arrangement of linear subspaces naturally arising from the action. Joint with Jack Jeffries.
 45pm: Anne Fahrner (Tübingen)
 Smooth Mori dream spaces with small Picard number
 In the case of toric varieties, it is well known that the projective spaces are the only smooth examples of Picard number one, Kleinschmidt gave a description of the smooth toric varieties having Picard number two and Batyrev studied the case of Picard number three. We extend these considerations in two directions: Firstly, we discuss (nontoric) rational varieties with a torus action of complexity one and secondly we look at intrinsic quadrics, i.e. varieties whose Cox ring is defined by a single homogeneous quadratic relation. In both cases we obtain explicit descriptions and, moreover, identify the Fano ones among the respective varieties.

 3 March 2017
 Leicester

 121pm: Gavin Brown (Warwick); Michael Atiyah Building MA119 (First Floor)
 Fanos in P^{2}xP^{2} 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 3folds, CalabiYau 3folds and 3folds of general type (the analogues of positively curved spheres, flat tori and hyperbolic gholed tori for g at least 2). Fano 3folds 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 3folds 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 3folds whose equations look like those of the Segre embedding of P^{2}xP^{2} (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 handson linear algebra. (This is in progress, and joint with Al Kasprzyk and Imran Qureshi.)
 23pm: 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 QGorenstein 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.
 3.304.30pm: 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

 12pm: 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 p^{2}+q^{2}+r^{2}pqr, where p,q,r are the weights of the arrows in the quiver. This is a joint work with Pavel Tumarkin.
 2.303.30pm: Ed Segal (Imperial)
 Homological projective duality for Pfaffians
 A Pfaffian variety is a space of antisymmetric 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 noncommutative resolutions, with the derived categories of projectivelydual pairs satisfying a nice relationship called "homological projective duality". I will explain what all of the above means, and describe the construction of these noncommutative 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 nonabelian GLSMs, proposed by Hori. This is joint work with Jorgen Rennemo.
 45pm: 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 twoorbit 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 Chevalleytype formula for the quantum cup product by the hyperplane class. Some of the varieties we consider have particularly wellbehaved 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

 121pm: 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 3folds, 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.
 23pm: 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 QQfactorial modifications of X. Via this construction we recover a Minkowski basis for the NewtonOkounkov bodies of Cartier divisors on X and hence generators of the movable cone of X.
 45pm: 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 BerensteinFominZelevinsky, 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.
 5.156.15pm: 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 LandauGinzburg models mirrordual to Fano manifolds. We give a mirror/symplectic account of this subject. This heavily utilizes the notion of an almosttoric 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 GrossSiebert.

 9 November 2016
 Nottingham, Physics C5

 12pm: 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.
 2.152.45pm: Ivan Cheltsov (Edinburgh)
 Del Pezzo surfaces, ample line bundles on them and their alphainvariants
 This "pretalk" is aimed at PhD students and nonexperts.
 34pm: Ivan Cheltsov (Edinburgh)
 Stable and unstable polarized del Pezzo surfaces
 YauTianDonaldson conjecture, recently proved by Chen, Donaldson and Sun, says that a Fano manifold is KahlerEinstein if and only if it is Kstable. Its stronger form, still open, says that a polarized manifold (M,L) is Kstable if and only if M admits a constant scalar curvature with Kahler class in L. In my talks, I will describe Kstability of ample line bundles on smooth del Pezzo surfaces (twodimensional Fano manifolds). I will show how to apply recent result of Dervan to prove Kstability and how to use flopversion of Ross and Thomas's obstruction to prove instability. This is joint work with Jesus MartinezGarcia.
 4.155.15pm: 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

 12pm: 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 CalabiYau threefolds admitting K3 fibrations. This talk is on joint work with Charles Doran, Andrew Harder, and Andrey Novoseltsev.
 2.303.30pm: 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.
 45pm: Miles Reid (Warwick)
 The TateOort group scheme of order p
 The aim is mostly group schemes in mixed characteristic, but the methods are mostly Galois theory of cyclotomic fields.

 1214 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

 12.301.30pm: Katrin Leschke (Leicester)
 Quaternionic Holomorphic Geometry
 In my talk, I will give a short introduction to Quaternionic Holomorphic Geometry: conformal maps into 3space 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.
 2.303.30pm: 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.
 45pm: 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

 23pm: Diane Maclagan (Warwick)
 Tropical ideals, varieties, and schemes
 3.304.30pm: Hendrik Suess (Manchester)
 Torus equivariant Kstability in complexity one

 5 February 2016
 Loughborough, Schofield Building

 1.302.30pm: Norbert Pintye (Loughborough)
 Complexity in Light of the CastelnuovoMumford Regularity
 34pm: 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 socalled 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

 23pm: 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 GuseinZade, Luengo and MelleHernández we study general abelian G, stating a conjectural product formula, and proving a homological stability result using a generalization of cores and quotients.
 3.304.30pm: Cristina Manolache (Imperial)
 Enumerative meaning of genus one GromovWitten 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 GromovWitten invariants of three folds.
 56pm: 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).
Travel
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:
 Alexander Kasprzyk
 School of Mathematical Sciences
 University of Nottingham
 University Park
 Nottingham
 NG7 2RD
Getting to Nottingham
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 3040 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.