CALF SEMINAR ABSTRACTS
This page contains abstracts from
previous Calf seminars. They are listed
alphabetically by speaker. For a listing by date, please return to
the main Calf page.
Tarig Abdel Gaidr (University of Glasgow)
[A^n /G] from the McKay quiver
Take G an abelian subgroup of SL(n,C) (n<4), it is well known
that the moduli space of certain representations of the McKay quiver
of G is isomorphic to GHilb (a crepant resolution of A^n/G). We will
use the McKay quiver to recover the stack [A^n/G] (a noncommutative
crepant resolution) and relate it to GHilb by a finite number of wall
crossings in some GIT chamber decomposition.
Mohammad Akhtar (Imperial College)
Mutations and Fano Varieties
This talk is an introduction to the theory of mutations. We will
discuss two closely related viewpoints on the subject: the algebraic approach
interprets mutations as birational transformations, and is concerned with their
action on Laurent polynomials. The combinatorial approach sees mutations as
operations on lattice polytopes, and allows one to construct deformations of the
corresponding toric varieties. We will explain the role played by algebraic
mutations in the program to classify Fano 4folds. We also discuss recent
results concerning combinatorial mutations of weighted projective surfaces.
The contents of this talk are joint work with Tom Coates, Alexander Kasprzyk and
Sergey Galkin.
Elizabeth Baldwin (University of Oxford)
Introduction to DeligneMumford Stacks; Parts I & II.
Stacks are a more general object than schemes. Their definition is very abstract; they are in fact categories,
and only after some work can their geometric structure be understood.
We will start part I by defining categories fibred in groupoids, with emphasis on examples and moduli
interpretations. From there we will move on to DeligneMumford stacks, and define the etale topology on these.
In part II we review the definition of categories fibered in groupoids. We then go on to look at how these may be
assigned geometric properties, via representable morphisms, and define DeligneMumford stacks.
Moduli of stable maps as a GIT quotient.
The moduli space of curves is one of the most fundamental objects in attempts to classify algebraic varieties or schemes. This may be generalised to the moduli space of pointed curves, and futher to moduli spaces of pointed maps (where the maps are from curves to a specified space).
Geometric invariant theory is a powerful tool for creating quotients in algebraic geometry. It is hence a way to construct moduli spaces. An early construction of the moduli space of curves was done in this way, by Gieseker. Along with David Swinaski, I have extended his method to construct the moduli space of pointed maps.
We will review the definitions of the spaces in question, and of geometric invariant theory, before going through an outline of the method.
Oliver E. Anderson (University of Liverpool)
Chow sheaves and hrepresentability.
Abstract: We start by giving an introduction to Suslin and Voevodsky's
theory of relative cycles and htopologies. After this is done we
briefly describe Kollar's theory of families of algebraic cycles
before proving that Suslin and Voevodsky?s presheaf of effective
relative cycles of dimension r and degree d is hrepresented by
Kollar?s coarse moduli space Chow_{r,d}(X/S). As an application we
give a modern proof of the following classical result: Let X be a
scheme over an algebraically closed field (of arbitrary
characteristic), then two cycles Z,Z' on X are algebraically
equivalent if and only if there is a cycle W such that Z + W and Z'
+ W are both positive and there passes an irreducible curve through
the corresponding points of the Chow scheme.
Gergely Berczi (University of Budapest)
Multidegrees of Singular Maps
Let G be a compact, semisimple Lie group. Given a affine algebraic variety in a complex vector space, invariant under
the linear Gaction on the vector space, one can ask for the multidegree of the variety, which is an element of the
equivariant cohomology ring of the vector space. This is a polynomial in dimT variables (T is the maximal torus),
which stores more information about the variety than the ordinary degree of the projective closure of it.
The goal is to show a toy example for calculating multidegrees coming
from singularities of maps, using equivariant localization, and show
how nonreductive quotients come into the picture.
Fabio Bernasconi (Imperial College)
Some remarks on singular Fano threefolds in positive characteristic
. According to the Minimal model Program, Fano varieties
constitute one of the building blocks of algebraic varieties. Over the
complex numbers we now have a good understanding of their behavior,
e.g. we know the vanishing of the cohomology groups of the structure
sheaf, that they are rationally connected varieties and, with some
assumptions on the singularities, that they are bounded. However their
geometry is still shrouded in mystery over fields of positive
characteristic. We will survey some of the known results, explaining
where difficulties and subtleties arise and I will discuss some of my
recent work aimed to bound their possible pathological behavior.
Alberto Besana (University of Milan)
Symplectic aspects of framed knots.
We propose an interpretation of the topological framing of a knotas a generating function for a Lagrangian
submanifold of a symplectic manifold; the setting is Brylinski's space of knots (embedding of S^1 in R^3) and
Maslov theory for Lagrangian submanifold. Examples and applications to existence of linear bundles with
prescribed curvature will be given.
Pawel Borowka (University of Bath)
Abelian surfaces and genus 4 curves.
I would like to present some basic facts on abelian varieties. In
particular I am interested in (1,3) polarised abelian surfaces. In
analogy with the theta divisor I will distinguish a curve in the
linear system of polarization.
Using an idea of Andreotti and Mayer I will prove that on a genaral
surface the resulting curve is smooth.
An easy exercise or an open problem?.
In my talk I will present the paper of H. Graf von Bothmer et. al.
(arXiv:math/0605090v2). Using a nice scheme theory they partially
proved the Casas Alvero question about polynomials in one variable.
Nonsimple abelian varieties
Abstract: In dimension two, the locus of nonsimple principally
polarised abelian varieties have infinitely many irreducible
components called Humbert surfaces. I will briefly explain the
situation there and show how to generalise the notion of Humbert
surface to higher dimensions to find irreducible components of the
locus of nonsimple principally polarised abelian varieties.
Nathan Broomhead (University of Bath)
Cohomology of line bundles on toric varieties.
One of the reasons for studying toric varieties is the ability to do certain computations simply, using the
combinatoric structure. In this talk we consider an example of this.
We recall with reference to P^{2} the usual construction of a toric variety from its fan. We then
introduce Cech cohomology, and use this to look at the cohomology of line bundles on toric varieties in terms
of cohomology calculated on the support of the fan. As an example of its use, we prove a vanishing theorem for
toric varieties.
The Dimer Model and CalabiYau Algebras.
From dimer models, first studied in physics, we can produce a class of CalabiYau algebras which are candidates for noncommutative crepant resolutions of Gorenstein 3fold affine toric singularities. In this talk I will introduce, via examples, dimers and their corresponding toric varieties. I will then talk briefly
about the "consistency" condition that underlies the CalabiYau property.
Tim Browning (University of Oxford)
Arithmetic of del Pezzo surfaces.
Del Pezzo surfaces provide beatiful examples of rational surfaces. Whilst their geometry is classical, they are
still somewhat myserious from the point of view of a number theorist. A basic property of such surfaces is that
there are infinitely many rational points on the surface provided there is at least one. I will discuss two
basic questions:
1) When does there exist a rational point?
2) Whenever the set of rational points is nonempty, how dense is it?
Jaroslaw Buczynski (University of Warsaw)
Linear sections of some Segre products..
I will explicitly describe and identify general linear sections of some
products of varieties (mainly P^1 \times P^n and P^1 \times \Q^{n1})
under their the Segre embeddings.
Legendrian varieties.
For given a vector space V with a symplectic form we define a subvariety
in P(V) to be legendrian if its affine cone has a Lagrangian tangent
space at each smooth point. We can prove that the ideal defining a
legendrian subvariety is a Lie subalgebra of the ring of functions on V
with Poisson bracket. Specially interesting is the case where the
ideal is generated by quadratic functions  then we can restrict our
considerations to a finite dimensional Lie algebra which happens to be
isomorphic to the symplectic algebra of V. Next we prove that the
subgroup of Sp(V) corresponding to the subvariety act transitively on
smooth points of the subvariety (in particular, if it is smooth then it
is homogeneous).
The next goal is fully classify legendrian subvarieties generated by
quadrics. There are not to many examples: twisted cubic in P^3, product
P^1 times Q_{n2} in P^{2n1} (where Q_{n2} is a smooth quadric in
P^{n1} and the embedding is Segre embedding) and four more exceptional
examples. This list appears in a paper of Landsberg and Manivel and also
in Mukai "Legendre varieties and simple Lie algebras". The conjecture is
that these are all possible smooth legendrian subvarieties generated by
quadrics. Moreover no singular example is known  so
possibly the assumption of smoothness is not necessary.
For more details see the preprint math.AG/0503528.
Vittoria Bussi (Oxford)
Categorification of DonaldsonThomas invariants and of Lagrangian intersections
We study the behaviour of perverse sheaves of vanishing cycles under action of symmetries and stabilization, and we investigate to what extent they depend on the function which defines them. We investigate the relation between perverse sheaves of vanishing cycles associated to isomorphic critical loci with their symmetric obstruction theories, pointing out the necessity for an extra "derived data". Similar results are proved for mixed Hodge modules and motivic Milnor fibres.
These results will be used to construct perverse sheaves and mixed Hodge modules on moduli schemes of stable coherent sheaves on CalabiYau 3folds equipped with ‘orientation data’, giving a categorification of DonaldsonThomas invariants. This will be a consequence of the more general fact that a quasismooth derived scheme with a (1)shifted symplectic structure and orientation data has a "categorification". Finally we categorify intersections of Lagrangians in a complex symplectic manifold, describing the relation with Fukaya categories and deformationquantization.
Paul Cadman (University of Warwick)
Deformations of singularities and the intersection form.
The nonsingular level manifolds of a miniversal deformation
of a singularity carry an intersection pairing in homology which can
be thought of geometrically as intersection of cycles. By a procedure
of Givental' and Varchenko it is possible to use a nondegenerate
intersection pairing to furnish the base space of the deformation with
a closed 2form. This is a symplectic form if the base space is even
dimensional. The symplectic form identifies a Lagrangian submanifold
in the discriminant of the deformation over which level sets share the
same degeneracy type.
I will explain this construction, give examples of the computation of
these symplectic forms and discuss the relationship between the
coefficients of the form and the equations of the Lagrangian
submanifold.
Francesca Carocci (Imperial College)
Homological projective duality and blow ups..
Kuznetsov's homological projective duality is a powerful tool for investigating semiorthogonal decompositions of algebraic varieties, which in turn are interesting as they seem to contain a lot of information about the geometry of the variety in question. I will recall the notion of homological projective duals and present a new example of geometric HP duals. Its construction is a special case of a more general story coming from blowing up base loci of linear systems. The example also highlights an interesting phenomenon: starting with a noncommutative HP dual pair one can obtain a commutative HP dual via the blowing up process. This example is a generalisation of other people's work on rationality of cubic fourfolds. This is joint work with Zak Turcinovic, Imperial College.
Gil Cavalcanti (University of Oxford)
Massey products in Symplectic Geometry.
For complex manifolds the "ddbar" lemma implies that the massey
products vanish and this is
how it is proved that Kaehler manifolds are formal. For symplectic manifolds satisfying the
Lefschetz property Merkulov proved a similar lemma using symplectic operators analogous to d
and dbar. The question that arises is "do Massey products vanish for such manifolds?"
I intend to give an example of symplectic manifold satisfying the Lefschetz property with
nonvanishing Massey products.
Notes for this talk are available.
Examples of generalized complex structures.
I'll introduce generalized complex structures and go through their basic properties, as
determined by Gualtieri in his thesis. I'll show how symplectic and complex structures
fit into this generalized setting. Then I'll present some results about existence of such
structures on manifolds that do not admit either complex or symplectic structures. I'll
also go through a classification of generalized structures on 6nilmanifolds and give
results about their moduli space.
Notes for this talk are available.
Andrew Chan (Warwick)
Gröbner Bases over Fields with Valuations
Gröbner bases have several nice properties that mean that certain problems in algebraic geometry can be reduced to the construction of a Gröbner basis. For example Gröbner bases allows us to easily determine whether a polynomial lives in some ideal, find the solutions to systems of polynomial equations, as well as having applications in robotics.
In this talk I shall introduce Gröbner bases and see the problems that arise when trying to adapt this theory to polynomial rings over fields with valuations. We shall discuss how these Gröbner bases are interesting to algebraic geometers and how they have important applications to tropical geometry.
Emily Cliff (Oxford)
Universal Dmodules
A universal Dmodule of dimension n is a rule assigning to every family of smooth ndimensional varieties a family of Dmodules, in a compatible way. This seems like a huge amount of data, but it turns out to be entirely determined by its value over a single formal disc. We begin by recalling (or perhaps introducing) the notion of a Dmodule, and proceed to define the category M_n of universal Dmodules. Following Beilinson and Drinfeld we define the GelfandKazhdan structure over a smooth variety (or family of varieties) of dimension n, and use it to build examples of universal Dmodules and to exhibit a correspondence between M_n and the category of modules over the groupscheme of continuous automorphisms of formal power series in n variables.
Giulio Codogni (Cambridge)
Curves, Jacobians and Modular Forms
In the first part of the talk, I will introduce the classical theory of Jacobians. In particular, I will stress the relations between the singularities of the Theta divisor and the projective geometry of the curve. In the second part, I will focus on the Schottky problem and on modular forms. I will discuss modular forms arising from lattices: these might provide upper bound on the slope of the moduli space of curves.
Alex Collins (University of Bath)
Representations of quivers and weighted projective lines.
Gaia Comaschi (Université des Sciences et Technologies de Lille 1)
Pfaffian representations of cubic threefolds and instanton
bundles.
Given a hypersurface X ⊂ P^n, we can determine whether its
equation might be expressed as the determinant of a matrix of linear
forms, by showing the existence of certain ACM sheaves on X. One of
the most ecient way to produce these sheaves is to use Serre's
correspondence, starting from AG subschemes of X. In this talk I
will treat the case where X is a cubic threefold. I will illustrate
how we can construct explicitly AG curves corresponding to Pfaffian
representations of X and how Serre's correspondence yields a
component of the moduli space of instanton bundles on X.
Barrie Cooper (University of Bath)
Koszul Duality and Twisted Group Algebras.
Let V be a representation of a finite group G. Then the symmetric (S) and exterior (L) algebras of V are Koszul dual
(over k), in the sense that S \otimes L^* is a bigraded algebra with a natural differential ofdegree (1,1) which is
exact (except in degree (0,0)). The exactness ofthe differential gives a wellknown recurrence for the symmetric
powers of a representation in terms of tensor products of exterior and symmetric powers. In particular, this gives a
recurrence on the McKay matrices of these representations.
In order to see how the matrix recurrence arises in a more direct way, we should consider the following:
Given a left kGmodule W, define a twisted bimodule structure on Twist(W) = W \otimes kG, where the right action is
(right) multiplication in kG and the left action is both the left action on W and (left) multiplication in kG. The
McKay matrix of W now coincides with its decomposition matrix in terms of the irreducible kGbimodules. Furthermore,
it can be shown that Twist(S) and Twist(L) are Koszul dual rings (over kG). Hence, the recurrence on the McKay
matrices reflects the fact that the differential respects the grading induced by the projectors onto the irreducible
kGbimodules.
We also discuss the related theory of almostKoszul rings and their connection with periodic recurrences.
An introduction to Derived Categories.
The derived category is a powerful homological tool and is the correct way of understanding derived
functors such as Tor and Ext. Many important papers in algebraic geometry and mathematical physics are
now being written in the language of derived categories, most notably Bridgeland, King and Reid's
"Mukai implies McKay: the McKay correspondence as an equivalence of derived categories".
We begin by discussing abelian categories, which satisfy precisely the axioms needed to define
homology. Next we encounter a problem for which homology isn't quite powerful enough to find a
solution  this leads naturally to the definition of the derived category of an abelian category. The
algebraic structure of the derived category is that of a triangulated category and we finish by trying
to understand the interplay between abelian, derived and triangulated categories.
McKay Matrices, CFT Graphs, and Koszul Duality (Part I).
To a finite subgroup of SL(n) we associate a graph. We explore the possibility of classifying such
graphs, and representation theory highlights a recurrence relation for which these graphs exhibit
unusual behaviour. We reduce the qualitative behaviour under this recurrence to a quantitative test,
which the rational conformal field theory graphs also appear to satisfy.
In subsequent talks we discuss how this test may betray the existence of a pair of Koszul or
almostKoszul dual algebras associated to the path algebra of the graph.
Stephen Coughlan (University of Warwick)
Introduction to graded rings and varieties.
Graded rings are a basic component in studying the birational geometry of algebraic surfaces and 3folds.
I illustrate graded rings and their relation to (weighted) projective space via the Proj construction.
I will go on to describe some algebraic varieties using graded ring methods.
Dougal Davis (LSGNT)
Some stacks of principal bundles over elliptic curves and their shifted symplectic geometry.
In a 2015 paper, I. Grojnowski and N. ShepherdBarron give a recipe which produces an algebraic variety from the ingredients of an elliptic curve E, a simple algebraic group G, and an unstable principal Gbundle on E. In the case where G = D_5, E_6, E_7 or E_8, they show that a particular choice of Gbundle yields a del Pezzo surface of the same type as G. It is an open question which varieties arise for different choices of Gbundle.
In this talk, I will describe how certain stacks of principal bundles on E, which are the main players in this construction, carry natural shifted symplectic and Lagrangian structures over the locus of semistable bundles. Time permitting, I will show how a very crude study of the degeneration of these structures at the unstable locus gives a much more direct computation of some of the canonical bundles appearing in the GrojnowskiShepherdBarron paper, which works for all groups and all bundles.
Ruadhaí Dervan (Cambridge)
An introduction to Kstability
A central problem in complex geometry is to find necessary and
sufficient conditions for the existence of a constant scalar curvature
Kahler metric on an ample line bundle. The YauTianDonaldson
conjecture states that this is equivalent to the algebrogeometric
notion of Kstability, related to geometric invariant theory. I will
give a gentle introduction to Kstability and time permitting there
will be some applications.
Carmelo Di Natale (Cambridge)
A period map for global derived stacks
In the sixties Griffiths constructed a holomorphic map, known as the local period map, which relates the classification of smooth projective varieties to the associated Hodge structures. Fiorenza and Manetti have recently described it in terms of Schlessinger's deformation functors and, together with Martinengo, have started to look at it in the context of Derived Deformation Theory. In this talk we propose a rigorous way to lift such an extended version of Griffiths period map to a morphism of derived deformation functors and use this to construct a period morphism for global derived stacks.
Will Donovan (Imperial College)
Tilting, derived categories and noncommutative algebras.
We can describe the derived categories of coherent sheaves on certain simple spaces by a method called tilting. This gives an equivalence of the derived category with another category built from a certain noncommutative algebra. We will work this out in some simple cases.
The McKay Correpsondence.
Kleinian surfaces singularities are obtained by taking a quotient of
the affine plane, under the action of a finite subgroup G of SL(2).
There exist certain minimal resolutions of such singularities: the
McKay correspondence tells us that the geometry of the resolution
remembers some of the representation theory of G, albeit in a subtle
manner. In particular, the irreducible components of the exceptional
locus turn out to be in bijection with nontrivial irreps of G. This
is part of a long (and continuing) story, bridging geometry and
algebra in deep ways, which I will not have time to go into. However I
will explain, following BridgelandKingReid, how derived categories
can give us an elegant insight into this correspondence.
Vivien Easson (University of Oxford)
Applying algebraic geometry to 3manifold topology.
In the study of 3dimensional manifolds, two of the most useful
structures to have are hyperbolic geometric structures and essential
surfaces lying in the manifold. Each of these relates to a different
kind of representation of the fundamental group. A series of papers
by Marc Culler and Peter Shalen has examined the interaction between
these, using some algebraic geometry of the character variety to
provide the connection. CullerShalen theory continues to provide new
insights and deep theorems in 3manifold topology. I intend to give an
accessible introduction to the various ideas involved.
Notes for this talk are available.
Vladimir Eremichev (University of Warwick)
GHilbert schemes and related constructions.
Let X be a complex quasiprojective variety and G a finite group
acting on X by automorphisms. The resulting quotient space X/G is
singular. In dimensions 2 and 3 we have a prefered crepant
(symplectic) resolution, namely the GHilbert scheme, defined by
Nakamura as the moduli space of Gclusters. In dimensions 4 and
higher the situation is more complicated  GHilb X usually fails
to resolve singularities and crepant resolutions exist only in very
special cases. In my talk I will introduce the GHilb, explain that
it is a resolution in lower dimensions and show what goes wrong in
higher dimensions, including possible ways of fixing it.
Daniel Evans (University of Liverpool)
Birationally rigid complete intersections.
In this talk I will introduce the method of maximal singularities
and how it is applied to prove birational (super)rigidity of Fano
varieties. In particular, I will consider the case of Fano complete
intersections of index one with certain singularities.
Andrea Fanelli (Imperial College London)
Lifting Theorems in Birational Geometry
In this talk, I will try to convince you of how important lifting pluricanonical sections is. Two main approaches can be used: the algebraic one, based on vanishing and injectivity theorems, and the analytic one, which relies on OhsawaTakegoshi type L^2extension theorems. Bypassing as much as possible the birational mumbo jumbo, I will eventually discuss the Dlt Extension Conjecture proposed by Demailly, Hacon and Păun.
Enrico Fatighenti (University of Warwick)
Hodge Theory via deformations of
affine cones
Hodge Theory and Deformation Theory are known to be closely related: many example of this phenomenon occurs in the literature, such as the theory of Variation of Hodge Structure or the Griffiths Residues Calculus.
In this talk we show in particular how part of the Hodge Theory of a smooth projective variety X with canonical bundle either ample, antiample or trivial can be reconstructed by looking at some specific graded component of the infinitesimal deformations module of its affine cone A.
In an attempt of a global reconstruction theorem we then move to the study of the Derived deformations of the (punctured) affine cone, showing how to find amongst them the missing Hodge spaces.
Aeran Fleming (University of Liverpool)
Kähler packings of projective, complex manifolds.
In this talk I will introduce the notion of Kähler packings and
explore their connections to multipoint Seshadri constants and
Nagata's conjecture. I will then briefly present a general strategy to
explicitly construct Kähler packings on projective, complex manifolds
and if time permits discuss some examples of blow ups of the complex
projective plane.
Joel Fine (Imperial College London)
Constant scalar curvature Kahler metrics on fibred complex surfaces.
I will spend half the talk motivating the search for constant scalar curvature Kahler metrics.
In particular I will explain why these special metrics should be of use in studying "the
majority of" smooth algebraic varieties (i.e. stably polarised ones). In the other half of
the talk I will explain how to use an analytic technique called an adiabatic limit to prove the
existence of constant scalar curvature Kahler metrics on a special type of complex surface.
This talk is based on the preprint math.DG/0401275.
Slides for this talk are available.
Peter Frenkel (Budapest University of Technology & Economics)
Fixed point data of finite groups acting on 3manifolds.
We consider fully effective orientationpreserving smooth actions of a
given finite group G on smooth, closed, oriented 3manifolds. We
investigate the relations that necessarily hold between the numbers of
fixed points of various noncyclic subgroups. All such relations are
in fact equations mod 2, and the number of independent equations
yields information concerning lowdimensional equivariant cobordism
groups. We determine all the equations for noncyclic subgroups G of
SO(3).
This talk is based on the preprint math.AT/0301159.
Pierre Guillot (University of Cambridge)
Algebraic cycles in the cohomology of finite groups. The
classifying space BG of an algebraic group G can be approximated by
algebraic varieties and therefore has a welldefined Chow ring CH^*BG,
which is useful in the study of varieties acted on by G.
Conjecturally this is the same as another ring defined topologically,
namely using complex cobordism. These rings come equipped with a
natural map to the ordinary cohomology ring. After explaining this in
some detail I will give some examples of computations, using tools
like the Steenrod algebra or the Morava Ktheories.
Eloïse Hamilton (University of Oxford)
Moduli spaces for Higgs bundles, semistable and unstable..
The aim of this talk is to describe the classification problem for
Higgs bundles and to explain how a combination of classical and
NonReductive Geometric Invariant Theory (GIT) might be used to obtain
moduli spaces for these objects.
I will start by defining Higgs bundles and explaining the
classification problem for Higgs bundles. This will involve
introducing the "stack" of Higgs bundles, a purely formal object which
allows us to consider all isomorphism classes of Higgs bundles at
once. Then, I will explain how this stack can be described
geometrically. As we will see, the stack of Higgs bundles can be
decomposed into disjoint strata, each consisting of Higgs bundles of a
given "instability type". I will explain how classical GIT can be used
to obtain a moduli space for the substack of semistable Higgs bundles,
and how nonreductive GIT might be applied to obtain moduli spaces for
the remaining unstable strata.
Umar Hayat (University of Warwick)
Gorenstein Quasihomogeneous Affine Varieties.
We study quasihomogeneous affine algebraic varieties, in particular their tangent bundle and canonical class, with the aim of characterising the case in which the variety is Gorenstein.
Thomas Hawes (University of Oxford)
GIT for nonreductive groups
Geometric invariant theory (GIT) is concerned with the question of constructing quotients of algebraic group actions within the category of varieties. This problem turns out to be sensitive to the kind of group being considered. When a reductive group G acts on a projective variety X, Mumford showed how to find an open subset X^s of X (depending on a linearisation of the action) that admits an honest orbit space variety X^s/G. Moreover, this admits a canonical compactification X//G, obtained by taking Proj of the finitely generated ring of invariant sections of the linearisation. This rather nice picture breaks down when the group G is not reductive, since there is the possibility of nonfinitely generated rings of invariants. This talk will look at work being done to describe a similar Mumfordstyle picture for nonreductive group actions. After reviewing Mumford's result for reductive groups, we will look at the work done by Doran and Kirwan on GIT for unipotent group actions, which provide the key for formulating GIT for general algebraic groups. We will finish by looking at work in progress on how to extend the ideas of Doran and Kirwan to the case where the group is not unipotent.
David Holmes (University of Warwick)
Jacobians of hyperelliptic curves.
Jacobians of curves are the natural higherdimensional analogues of
elliptic curves, and many of the familiar properties of elliptic
curves carry over. In particular, the Mordell Weil theorem (that the
group of rational points over a number field is finitely generated)
holds on any Jacobian, and the proof is again based on a theory of
heights.
After giving basic definitions, we will look at how to use this to
find an algorithm to compute the torsion part of the MordellWeil
group of the Jacobian of a hyperelliptic curve, giving a method to
explicitly construct the Jacobian and exploring why this isn't enough.
Julian Holstein (University of Cambridge)
Preserving K(pi,1)'s  Hyperplane arrangements and homotopy type.
Katzarkov, Pantev and Toen define schematic homotopy types as algebraic models for topological spaces. In this talk I will look at some properties of their construction in the case of hyperplane arrangements.
Vicky Hoskins (University of Oxford)
An introduction to stacks.
Stacks are needed to give a geometric space to moduli functors that are not
representable by a scheme, e.g. M_g the stack of smooth curves of genus g. In
this sense we can see stacks as generalisations of schemes. In this talk we
approach stacks from two different viewpoints. Firstly we view them as
pseudofunctors from the category of schemes to the 2category of groupoids; this
point of view originates from our motivation, moduli functors. Secondly we
describe the slightly more common definition, using categories fibred over
groupoids. The aim is by the end of the talk to give the definition of a
DeligneMumford stack and also give some examples.
Daniel Hoyt (University of Cardiff)
Braided categories and TQFTs.
Topological quantum field theories (TQFTs) have proven an interesting tool in topology, providing
invariants of 3manifolds; to every (threedimensional) TQFT there is a
"quantum invariant". But how does one construct a TQFT? One solution is
through using categories with extra structure, such as a tensor product. In this talk I
plan to define both TQFTs and an important class of category (braided
categories) that can be used in the construction of TQFTs. I will also give a few
examples that demonstrate how familiar braided categories really are.
Anton Isopoussu (Cambridge)
Kstability, convex cones and fibrations
Test configurations are a basic object in the study of canonical metrics and Kstability. We introduce two ideas into the theory. We extend the convex structure on the ample cone to the set of test configurations. The asymptotics of a filtration are described by a convex transform on the Okounkov body of a polarisation. We describe how these convex transforms change under a convex combination of test configurations.
We also discuss the Kstability of varieties which have a natural projection to a base variety. Our construction appears to unify several known examples into a single framework where we can roughly classify degenerations of fibrations into three different types: degenerations of the cocycle, degenerations of the general fibre and degenerations of the base.
SeungJo Jung (Warwick)
Moduli of representations of McKay quiver
This talk describes representations of McKay quiver and moduli spaces of them. Specially, for a finite group A in SL(3), I introduce AHilbC^3 in terms of moduli space of McKay quiver representations. If time permits, we can discuss moduli spaces of McKay quiver for a finite groups in GL(3).
AnneSophie Kaloghiros (Cambridge)
The defect of terminal quartic 3folds.
Let $X \subset \mathbb{P}4$ be a quartic 3fold with terminal singularities. The Grothendieck Lefschetz
theorem states that any Cartier divisor on X is the restriction of a Cartier divisor on $\mathbb{P}4$.
However, no such result holds for Weil divisor. If the quartic X is not assumed to be $\mathbb{Q}$factorial,
very little is known about its group of Weil divisors. $\mathbb{Q}$factoriality is a global topological property,
and very ''simple" quartics fail to be $\mathbb{Q}$factorial.
More generally, one could consider Gorenstein terminal Fano 3folds of Picard rank 1. Can one bound the
rank of the group of Weil divisors of a terminal Gorenstein quartic (Fano) 3fold of Picard rank 1?
I will give such a bound for quartics and for some Fanos. I will also show that if a quartic is not
$\mathbb{Q}$factorial, then it contains a (Weil non Cartier) surface of low degree. There is a finite
number of possibilities for these surfaces.
Grzegorz Kapustka and Michal Kapustka (Jagiellonian University, Krakow)
Some geometric properties of singular del Pezzo surfaces.
In this talk we study geometric properties of singular del Pezzo surfaces with log terminal singularities of
index less then or equal to 2. We study their (in some way) canonical embedding and use it to describe them
with equations in some weighted projective space.
Grzegorz Kapustka (Jagiellonian University, Krakow)
Linear systems on an Enriques surface.
The aim of the talk is to describe linear systems of an irreducible curve on an Enriques surface, and the
maps associated with these linear systems. We ask when a map is a morphism, what is the degree of this
morphism, and describe the eventual singularities by looking at the image.
Michal Kapustka (Jagiellonian University, Krakow)
Linear Systems on a K3 surface.
The aim of this talk is to describe linear systems on K3 surfaces. We are mostly concerned with their base
points (or components), the morphism associated with them and its image. We also try to introduce the notion
of Seshadri constants and we show some examples of linear systems on some K3 surface where we can compute
them.
Alexander Kasprzyk (University of Bath)
Introduction to Toric Varieties.
Toric varieties form an important class of algebraic varieties whose particular strength lies in methods of
construction via combinatorial data. Understanding this construction has led to the development of a rich
dictionary allowing combinatorial statements to be translated into algebraic statements, and vice versa.
In the first talk the basic details of the combinatorial approach to constructing toric varieties are given.
The construction is motivated by specific examples from which the more general methods can be deduced.
The second talk will concentrate on the torus action on the variety. We will discuss the orbit closure and
introduce the "star" construction. Finally, we shall apply what we have learnt to toric surfaces,
analysing the singularities and seeing how they are resolved.
Notes for the first talk are available.
Recognising toric Fano singularities.
It is a well known fact that toric Fano varieties of dimension n correspond to convex polytopes in R^n. In
particular, if the variety has at worst terminal singularities, then the associated polytope is a lattice
polytope P such that P\cap Z^n consists of the vertices of P and the origin. A similar condition on the
polytope exists when the variety is allowed to possess canonical singularities. In this talk I intend to
review the definitions of the singularities involved, and hopefully shed some light on these equivalencies.
Some basic knowledge of toric geometry will be assumed.
What little I know about Fake Weighted Projective Space.
At the December 2003 Calf in Warwick, Weronika Krych introduced me to
the idea of fake (or false) weighted projective space. These are objects which arise naturally in the context
of toric geometry, and are quotients of bona fide weighted projective space. Fake weighted projective spaces
also arise in toric Mori theory. Loosely speaking, they appear as the fibres of an elementary contraction.
We shall see that a great deal of information about the singularities of a fake weighted projective space can
be deduced from weighted projective space. We shall also establish bounds on how ``far away'' these fake weighted
projective spaces can be from weighted projective space whilst still remaining sufficiently ``nice''.
Jonathan Kirby (University of Oxford)
Model Theory and Geometry  An Introduction.
The talk will be in two parts. In the first I will explain what model theory is, and how it can be
thought of as a generalization of the study of zeros of polynomials (aka Algebraic Geometry). In the
second I will explain how simple geometric ideas crop up naturally in model theory. The aim is to give
an overview of the ideas rather than any technicalities, and no familiarity with logic will be
assumed.
Weronika Krych (University of Warsaw)
False weighted projective spaces and Mori theorem for orbifolds.
We define false weighted projective spaces as toric varieties with fan constructed from vectors v_0,...,v_n in
lattice Z^n with sum_{i=0}^n (a_i * v_i) = 0 for some integers a_i.
The only difference of this fan and the one of weighted projective space is that the v_i's do not span the lattice.
False weighted projective space are quotients of P(a_0,...,a_n) by the action of a finite group.
We distinguish false ones by introducing the fundamental group in codimension 1 and proving it is nontrivial
exactly for false ones.
False projective spaces are quotients of P^n and they are orbifolds.
We conjecture a generalization of Mori theorem characterizing P^n as the only projective varieties with ample
tangent bundle.
Roberto Laface (Leibniz Universität Hannover)
Decompositions of
singular Abelian surfaces
Inspired by a work of Ma, in which he counts the number of decompositions of abelian surfaces by latticetheoretical tools, we explicitly find all such decompositions in the case of singular abelian surfaces. This is done by computing the transcendental lattice of products of isogenous elliptic curves with complex multiplication, generalizing a technique of Shioda and Mitani, and by studying the action of a certain class group act on the factors of a given decomposition. Incidentally, our construction provides us with an alternative and simpler formula for the number of decompositions, which is obtained via an enumeration argument. Also, we give an application of this result to singular K3 surfaces.
Marco Lo Giudice (University of Bath, and University of Milan)
Schemetheoretic projective geometry.
We will introduce projective geometry in the language of schemes. Starting from the projective spectrum of a graded ring
we will explain some basic properties of projective varieties.
Introduction to schemes.
People tend to think about "schemes" to be synonymous with "Algebraic
Geometry" but this is not quite true. As a result learning the machinery can be really
frustrating, as our geometric intuition doesn't seem to fit into the picture. Actually the
theory of schemes is far more general than Algebraic Geometry, and many concepts arising in
the geometric context make sense only for a particular kind of schemes usually called
"algebraic schemes". I will define algebraic varieties from this point of view,
avoiding too much abstract nonsense and retaining the geometric point of view in evidence.
Detailed notes on scheme theory are
available.
Artin level
algebras.
Artin level algebras are zero dimensional graded algebras, they are a generalization of Gorenstein algebras.
I will describe their Hilbert function and their graded minimal resolution.
Cormac Long (University of Southampton)
Some results on Coxeter groups.
We give necessary conditions for the {3,5,3} Coxeter group to surject onto PSL(2,p^n). We also look at some of
the manifolds arising from the low index normal subgroups of this group.
Andrew MacPherson (Imperial College London)
Mirror Symmetry is Tduality
The SYZ conjecture suggests that mirror manifolds should admit fibrations by dual special Lagrangian tori. I'll talk about some of the motivation for, and consequences of, this conjecture, and then I'll say something about what some people are trying to do about it. Be warned that this talk will be both i) very imprecise and ii) not particularly algebraic.
A nonarchimedean analogue of the SYZ conjecture
The SYZ conjecture is a statement, or rather, a framework of statements, about the geometry of the large complex structure and large radius limit points of the moduli space of CY nfolds, and the mirror involution that exchanges them. Following proposals of Kontsevich, I'll talk about how nonArchimedean geometry can be used to study this limit in a more algebrogeometric setting.
Diletta Martinelli (Imperial College London)
Semiampleness of line bundles in positive characteristic
I will explain why the property of semiampleness is very important in algebraic geometry and I will present some sufficient conditions for the semiampleness of a line bundle on a variety defined over the algebraic closure of a finite fields. In the second part of the talk I will present some results that are part of a joint work with Jakub Witaszek and Yusuke Nakamura.
Mirko Mauri (LSGNT)
Dual complexes of log CalabiYau pairs and Mori fibre spaces.
Dual complexes are CWcomplexes, encoding the combinatorial data of
how the irreducible components of a simple normal crossing pair
intersect. They have been finding useful applications for instance in
the study of degenerations of projective varieties, mirror symmetry
and nonabelian Hodge theory. In particular, Kollár and Xu conjectures
that the dual complex of a log CalabiYau pair should be a sphere or a
finite quotient of a sphere. It is natural to ask whether the
conjecture holds on the end products of minimal model programs. In
this talk, we will validate the conjecture for Mori fibre spaces of
Picard rank two.
Francesco Meazzini (Sapienza Università di Roma)
QUIVER REPRESENTATIONS AND GORENSTEINPROJECTIVE MODULES.
We consider a finite acyclic quiver Q and a quasiFrobenius ring R. We then characterise Gorensteinprojective modules over the path algebra RQ in terms of the corresponding quiver representations over R, generalising the work of X.H. Luo and P. Zhang to the case of not necessarily finitely generated RQmodules.
We recover the stable category of Gorensteinprojective RQmodules as the homotopy category of a certain model structure on quiver representations over R.
Carl McTague (University of Cambridge)
The Cayley plane genus.
I will give a new geometric characterization of the Witten genus.
Jasbir Nagi (University of Cambridge)
Graded Riemann spheres.
Riemann spheres are extremely useful in the study of twodimensional conformal field theories.
One can ask what is the corresponding structure to look at if one wishes to study a superconformal field theory.
One way of introducing anticommuting coordinates is to consider the sheaf functions on the Riemann sphere, and
extend them by
anticommuting variables.
This can be more useful than a superspace formalism, since there is still a notion of a "patching
function" on intersections of "coordinate patches".
This talk is based on the preprint hepth/0309243.
Ciaran Meachan (University of Edinburgh)
Moduli of Bridgelandstable objects.
In the spirit of Arcara & Bertram, we investigate wallcrossing phenomena in the stability manifold of an irreducible principally polarized abelian surface for objects with the same invariants as (twists of) ideal sheaves of points. In particular, we construct a sequence of fine moduli spaces which are related by Mukai flops and observe that the stability of these objects is completely determined by the configuration of points. Finally, we use FourierMukai theory to show that these moduli are projective.
Oliver Nash (University of Oxford)
An Introduction to Twistor Theory.
An introduction to the Penrose twistor corresponce will be presented. We will begin by discussing
the correspondence between conformal fourmanifolds and appropriate complex threemanifolds. In
particular, our discussion will include the usual Penrose transform in this case. We will then
discuss the various generalisations of the correspondence to other dimensions and geometric
structures. We will conclude by describing some of the applications of twistor theory to gauge
theory (monopoles and instantons), existence of complex structures and deformations of hypercomplex
structures.
Igor Netay (HSE, Moscow)
On Ainfinity algebras of highest weight orbits
I will present recent results on syzygy algebras. For any algebraic variety X > P^n with an embedding to projective space the syzygy spaces have a natural structure of an Ainfinity algebra. I will discuss the case of projectivization of highest weight orbits in irreducible representations of reductive groups.
Alvaro Nolla de Celis (University of Warwick)
Introduction to cyclic quotient singularities.
I will introduce quotient singularities and their resolution, in particular I will talk about Du Val
singularities or rational double points, giving a descriptions of their resolution in terms of HirzebruchJung
continued fractions and Dynkin diagrams.
Claudio Onorati (University of Bath)
Moduli spaces of generalised Kummer varieties are not connected
Using the recent computation of the monodromy group of irreducible holomorphic symplectic (IHS) manifolds deformation equivalent to generalised Kummer varieties, we count the number of connected components of the moduli space of both marked and polarised such manifolds. After recalling basic facts about IHS manifolds, their moduli spaces and parallel transport operators, we show how to construct a monodromy invariant which translates this problem in a combinatorial one and eventually solve this last problem.
John Christian Ottem (University of Cambridge)
Ample subschemes
We discuss how various notions of positivity of vector bundles is related to the geometry of subschemes.
Asymptotic cohomological functions.
Asymptotic cohomological functions were introduced by Demailly and Küronya to measure the growth rate of the cohomology of high tensor powers of a line bundle L. These functions generalize the volume function of a line bundle and capture a lot of the positivity properties of L. In this talk I will review some recent results on them by Demailly, Küronya and Matsumura and explain how they compare with other notions of weaker positivity of a line bundle.
Kyriakos Papadopoulos (University of Liverpool)
Reflection Groups, Generalised Cartan Matrices & KacMoody Algebras.
This talk will be the continuation of my talk in the Calf seminar in Liverpool (January 2005). I will spend a few
minutes talking about reflection groups in integral hyperbolic lattices, and use this machinery to define the
geometric realisation of a generalised Cartan matrix. There will be a short introduction to infinitedimensional Lie
algebras, based on the theory that we will give for generalised Cartan matrices.
Notes for this talk are available.
Reflection groups of integral hyperbolic lattices.
This is an introductory talk on reflection groups of integral hyperbolic symmetric bilinear forms.
Lobachevskii (hyperbolic) geometry is a strong tool in mathematics, and lots of problems which appeared in
algebraic geometry have been attacked using this tool. We will divide the lecture into two parts; in the
first one we will present all the preliminaries, and in the second part we will formulate Vinberg's
algorithm. This algorithm permits us to find all cells of a polyhedron C of an acceptable set P(C) of
orthogonal vectors to C, where C is the fundamental chamber for a subgroup W of the group W(M) (the group
generated by reflections in all elements of M), where S:MxM > Z is a given quadratic form. Hopefully,
this material will be used as a basis, for a future lecture on hyperbolic KacMoody algebras.
Andrea Petracci (Imperial College London)
On the quantum periods of del Pezzo surfaces
I will discuss a conjecture, due to Coates, Corti, Kasprzyk et al., which relates the quantum cohomology of del Pezzo surfaces with isolated cyclic quotient singularities to combinatorial data coming from lattice polygons and Laurent polynomials. I will present evidence for this conjecture in the case of del Pezzo surfaces with 1/3(1,1) singularities. The ideas discussed are in the spirit of recent work by CoatesCortiGalkinKasprzyk, who used quantum cohomology to reproduce the IskovskikhMoriMukai classification of smooth Fano 3folds. This is joint work with A. Oneto.
Matthew Pressland (University of Bath)
Labelled Seeds and Mutation Groups
This talk will introduce labelled seeds, whose definition is a modification of that of seeds of a cluster algebra. Under this new definition, the cluster algebra itself will be unchanged, but the set of labelled seeds will form a homogeneous space for a a group of mutations and permutations. We will study the automorphism group of this space, and conclude that for certain mutation classes, the orbits of this automorphism group consist of seeds with "the same cluster combinatorics", in the sense that their quivers are all related by opposing some connected components. Knowledge of cluster algebras will not be assumed, and indeed one goal is to provide an introduction to the subject, albeit in a slightly esoteric way.
Ice Quivers with Potential and Internally 3CY Algebras.
A dimer model, which is a bipartite graph on a closed orientable surface, gives rise to a Jacobian algebra. Under consistency conditions on the dimer model, this algebra satisfies a very strong symmetry condition; it is 3CalabiYau. However, the consistency condition forces the surface to be a torus. This can be avoided by allowing surfaces with boundary, on which dimer models give rise to frozen Jacobian algebras. We define a suitable modification of the 3CalabiYau property for these algebras, and explain some interesting clustertheoretic results that follow from it.
Thomas Prince (Cambridge)
From scattering diagrams to GromovWitten theory
This talk will be a survey of the paper of Gross, Pandharipande and Siebert on enumerative consequences of their scattering diagram calculations. In particular recalling the notions of scattering diagrams, tropical curves and the KontsevichSoibelman lemma before discussing the holomorphic analogues of the tropical curve counts. This is also supposed to be valuable background material for reading recent papers of Gross Hacking and Keel on mirror symmetry for log CalabiYaus.
Qiu Yu (University of Bath)
Stability space of quivers/species of two vertices
I'm going to describe the stability space (in the sense of Bridgeland) of the quiver A_2. As a comparison, I will show that this space 'contains' the fundamental domain of stability space of the Kronecker quiver P_2 (or equivalently, of the projective space of dimension 1) in some sense. Then I will explain the folding techniques to describe the stability space of the species of type B_2=C_2 and G_2.
Lisema Rammea (University of Bath)
Some Interesting Surfaces of General Type in Projective 4space
A well known theorem of Gieseker says that there exists a
quasiprojective coarse moduli scheme for canonical models of surfaces
of general type S with fixed K^{2}_{S} and c_{2}(S). However there are
some classical inequalities which a surface of general type must
satisfy. Beyond these numerical restrictions the study of surfaces of
general type largely consist of studying examples in : (1)
"Geography"deciding which Chern numbers or other topological
invariants arise as the invariants of a minimal surface of general
type, and
(2)"Botany"for decribing all the deformation types within a fixed
topological type. In this talk we look at (1).
Construction of NonGeneral Type surfaces in P^4_w.
We wish to generate smooth NonGeneral Type surfaces in four
dimensional weighted projective space, P^{4}_w. For trivial
weights (all weights equal to 1), a lot of work has been done by
various people. In this case it is known that all surfaces of degree
greater or equal to 52 are of general type. The conjectured bound is
15. Decker et al generated examples of smooth NonGeneral Type
surfaces using an earlier version of the computer algebra system
Macaulay2 in the case of trivial weights. We study their construction
methods to try and come up with an efficient method to generate
NonGeneral Type surfaces in P^{4}_w, where not all the
weights equal one. For now we insist that our weights are pairwise
coprime. Nontrivial weights lead naturally to cyclic quotient
singularities. Examples of K3 surfaces have been found by Altinok et
al in P^{4}_w. We discuss construction of an Enriques' surface
in P^{4}_w by taking an example with w=(1,1,1,1,2).
Nils Henry Rasmussen (University of Bergen)
The dimension of W^1_d(C) where C is a smooth curve on a K3 surface
Jorgen Rennemo (Imperial College)
Gottsche's ExConjecture and the Hilbert Scheme of Points on a Surface
Consider a smooth, projective surface with a line bundle L on it. We say a curve is dnodal if it has d singular points that are nodes and no other singularities. The Göttsche Conjecture (now a theorem) is a statement about the number of dnodal curves in a ddimensional linear system of divisors of class L. The first aim of this talk will be to explain this statement in some detail. I will then introduce the Hilbert scheme of points on a surface and show how the conjecture can be reduced to the computation of a cohomology class on the Hilbert scheme. This is the first step in one of the known proofs of the conjecture.
Sönke Rollenske (Imperial College London)
Some very nonKahler manifolds.
In the first part of the talk I want to give an elementary solution to the
classical question how much de Rham and Dolbeault cohomology can differ on a
compact complex manifold (cf. [GriffithHarris78], p.444).
In the second part I will explain how this fits into the more general
framework of nilmanifolds with leftinvariant complex structures and how
these can be used to produce manifolds with interesting properties.
(reference: arXiv:0709.0481)
Taro Sano (University of Warwick)
Deformation theoretic approach to the classification of singular Fano 3folds.
Smooth Fano 3folds are classified classically and there are around 100 different families of them. If I allow terminal singularities on Fano 3folds, things get much complicated and the classification is not completed. I will explain difficulties in the classification of those Fano 3folds and how to make the classification easier by considering their deformations.
Deformations of weak Fano manifolds
The Kuranishi space of a projective variety is the parameter
space of small deformations of the variety.
It is important in the study of moduli spaces of projective varieties.
In many cases, the Kuranishi space is singular.
However, it is smooth in some important cases.
I will explain when the smoothness holds.
Shu Sasaki (Imperial College London)
Crystalline cohomology and crystals.
Crystalline cohomology originated from the observation that ladic cohomology groups of a smooth projective
(connected) variety over an algebraically closed field of characteristic p=l are "miserable" in comparison to
p\neq l case. Roughly speaking, Grothendieck's idea (outlined in his lectures at IHES in 1966) was to lift varieties to
characteristic zero and then take the de Rham cohomology to obtain "nice" (padic) cohomology. However, still
some questions remained. Most notably: Is it always the case that one can lift varieties? To remedy this situation, we
needed more sophisticated and subtle theory. The answer was... the theory of crystalline cohomology! In my talk, I'd
like to explain why this crystalline cohomology is the "right" one and if time permitting, I would hope to
talk about things like Fcrystals to illustrate how mindboggling this theory can sometimes be. I shall start from very
basics such as Grothendieck topology so don't be scared of what I've
just said above.
Danny Scarponi (Oxford/Tolouse)
The degree zero part of the motivic polylogarithm and the
DeligneBeilinson cohomology..
Last year, G. Kings and D. R ̈ossler related the degree zero part of the
on abelian schemes pol0 with another object previously defined by V. Maillot and D.
R ̈ossler. More precisely, the canonical class of currents constructed by Maillot and
R ̈ossler provides us with the realization of pol0 in analytic Deligne cohomology.
I will show that, adding some properness conditions, it is possible to give a
refinement of Kings and R ̈ossler’s result involving DeligneBeilinson
instead of analytic Deligne cohomology.
Ed Segal (Imperial College London)
Operads and the Moduli of Curves.
This talk will be an (attempted) explanation of Kevin Costello's paper math.AG/0402015.
I'll go through the definition of Ainfinity algebras and show how the universal structure
(operad) describing them relates to moduli spaces of riemann surfaces with boundary. We'll see that up to
homotopy equivalence, these moduli spaces have a simple combinatorial description.
Crepant resolutions and quiver algebras
A resolution of a singularity is called 'crepant' if it's canonical bundle is trivial. For some singularities it's possible to find a
noncommutative algebra A, which we can draw as a quiver,
such that modules over A are 'the same' as sheaves on a crepant resolution of the singularity (the derived categories are equivalent). In Van den Bergh's
terminology A is a 'noncommutative resolution'. I'll describe the ways that this can be done and discuss the various interpretations of the resulting quivers and their representations. If time permits I might explain the conjectural significance of A_infinity deformation theory in this context.
Despite some the hightech material in the above paragraph, most of this talk will be about a simple example.
Superpotential algebras from threefold singularities.
The orbifold X = C^3 / Z_3 is a simple but interesting example of a (noncompact) CalabiYau threefold. Physicists predict
that type II string theory on X reduces in the lowenergy limit to a gauge theory, which is described by a quiver and a superpotential. We'll discuss how these objects arise mathematically.
Lars Sektnan (Imperial College)
Algebrogeometric obstructions to the existence of cscK metrics on toric varieties.
The existence of constant scalar curvature (cscK) metrics on Kähler manifolds is a central problem in Kähler geometry. There are several known obstructions to the existence of such metrics and the algebrogeometric notion of Kstability is conjectured to be equivalent to this. We will present a classical obstruction, the Futaki invariant, in the toric setting and use it to show that the blowup of P^2 with its anticanonical polarisation does not admit a cscK metric. We will then show that this is not enough, by exhibiting an example due to WangZhou of a toric variety with vanishing Futaki invariant, which is not Kstable. Along the way we will introduce filtrations of the homogeneous coordinate ring of a polarised projective variety and discuss how these relate to Kstability and also give a stronger stability criterion. I will begin with a reminder on toric geometry.
Yuhi Sekiya (University of Nagoya)
Moduli spaces of McKay quiver representations.
The derived category of the minimal resolution of a
Kleinian singularity is equivalent to the derived category of a
certain noncommutative algebra. I will illustrate that the minimal
resolution is recovered as a moduli space of modules over the non
commutative algebra.
Michael Selig (Warwick University)
Orbifold RiemannRoch in high dimensions.
We are interested in explicit constructions of 3folds and 4folds with given
invariants.
We use the following wellknown graded ring construction: given a polarised variety
(X,D), under certain assumptions the graded ring R(X,D) = ⊕
_{n≥0}H^{0}(X,nD) gives
an embedding XProj(R(X,D)) ∈ wℙ. It is well known that the numerical data of
(X,D) is encoded in the Hilbert series P_{X}(t) := ∑
_{n≥0}h^{0}(X,nD)t^{n}.
We aim to break down the Hilbert series into terms associated to the orbifold loci
of X.
The talk should be fairly introductory. I will explain the ideas behind the work
from scratch, exhibit some results in 3D and explain some ideas for the 4D case.
Orbifold RiemannRoch and Hilbert Series.
Given a polarised orbifold (X,D) and its associated graded ring R = R(X,D) its numerical invariants (such as the plurigenera and the singularity basket) are encoded in its Hilbert series P_X(t). Studying the Hilbert series is therefore a sensible thing, as we could hope to use it to find generators and relations for the graded ring R. We deconstruct the Hilbert series into a sum of terms where each term corresponds clearly to an orbifold locus; using these methods we find a similar more general deconstruction of rational functions with poles only at roots of unity.
Kenneth Shackleton (University of Southampton)
Tightness and Computing Distances in the Curve Complex.
We give explicit bounds on the intersection number between any curve on a tight geodesic and the two ending curves.
We use this to construct all tight geodesics and so conclude that distances are computable. The algorithm applies
to all surfaces. The central argument makes no use of the geometric limit arguments seen in the recent work of
Bowditch (2003) and MasurMinsky (2000). From this we recover the finiteness result of MasurMinsky for tight geodesics.
This talk is based on the preprint math.GT/0412078.
Alexander Shannon (University of Cambridge)
Twistor Dmodules.
A desire to extend Hodge theory to ever more general general geometric settings necessitates a corresponding generalisation in the structures we use to describe it. I will review some aspects of Saito's theory of Hodge modules, which play the role of sheaves of Hodge structures on varieties, and give an indication of how the algebraic data can be recast in a more geometric way to give the more flexible twistor Dmodules of Sabbah.
Geometry without geometry.
We all know how to compute the (topological) cohomology of
an elliptic curve in various standard ways, but let's pretend we've
forgotten, and all we know about is a small piece of the derived
category of coherent sheaves (I'll start with a reminder of what this
is), but large enough that it generates the whole thing. Then we can
get the answer purely algebraically, along with the Hodge structure
and its variation with the parameter defining the elliptic curve, by
looking at structures on the cyclic homology of what turns out to be a
fairly small (and thus easy to work with explicitly) dg category.
Time permitting, I shall try to suggest why this is a potentially
interesting point of view for exploring ways of how elliptic curves
might degenerate in the world of noncommutative geometry.
Dirk Schlueter (University of Oxford)
DM stacks in toric geometry and moduli theory
This talk will be a followup to last term's introduction to stacks. The aim
will be to show algebraic stacks in action: as a first example, I will discuss
weighted projective spaces and toric geometry from the point of view of Deligne
Mumford stacks. The second part of the talk will focus on how algebraic stacks
come up in moduli problems and in what sense they record more information than
the classical coarse moduli schemes. As a guiding example, I will discuss
moduli spaces of (marked) curves and some of the maps between them.
YongJoo Shin (Sogang University)
Classification of involutions on a surface of general type with p_g=q=0
We would like to understand involutions on a minimal
surface of general type with p_g=q=0. Especially, for the surface
with K^2=7 we give a table classified branched divisors and
birational models of the quotient surface induced by an involution.
And we explain how to get the table, and which cases are supported by
examples.
James Smith (University of Warwick)
Introduction to K3 surfaces.
The two talks will cover some basic aspects of K3 surfaces with the following aims:
First, to say what a K3 surface is and how to recognise one. Examples will be given and the
difficult question of why K3s are interesting may be tackled.
Second, to become familiar with some of the methods used in the study of K3 surfaces such as
lattice theory and Hodge structures. Time permitting, we may look at some deeper aspects of the
subject and try and build an understanding of the moduli space of K3 surfaces.
In the second talk, by looking at explicit examples, we shall illustrate some general properties of K3 surfaces. In
particular, we look at variations of Hodge structure, periods and the associated PicardFuchs differential equation, and
use these to visualise the moduli space of certain oneparameter families of K3 surfaces.
Notes for the second part of this talk are available.
K3s as quotients of symmetric surfaces.
We consider the action of finite subgroups of SO(4) on P^3. Recent work of W. Barth and A. Sarti provides
three examples of families of K3 surfaces that arise as the quotient of invariant surfaces modulo this
group action. We describe an easy way to prove this and to find more examples using graded ring methods
and invariant theory. This talk will cover a basic introduction to algebraic K3 surfaces and will
demonstrate the use of graded rings and weighted projective spaces to their study.
David Stern (University of Sheffield)
Tilting Tstructures, Mutating Exceptional Collections, Seiberg Duality... It's all quivers to me.
In this I will working in the context of the bounded derived categories of coherent sheaves of a fano surface Z
and it's canonical bundle \omega_{Z} which is a Calabi Yau 3fold. I will breifly state how quivers relate to
tstructures and tilting them, to exceptional collections and their mutations, and if any physists are present
to seiberg duality. I will then use this to explain Tom Bridgeland's result in "Tstructures on some local
CalabiYau varieties" and if all goes well give a brief description of my current work.
Vocabulary made easy.
The aim of this talk is to provide an alternative understanding of derived categories, focusing on using the
formal definitions of a tstructure and the heart of a tstructure to get visual understanding of what a derived
category is, even for those with little prior knowedge. Then depending on time and peoples interest I will use
this 'picture' to give simple explainations, of things like torsion pairs, tilting with respect to torsion
pairs, stability conditions (Tom Bridgelands description), etc.
Jacopo Stoppa (Imperial College)
Stability and blowups.
We show that K and and Chow stability of the blowup of some
polystable variety along a 0cycle is related to the Chow stability of the
cycle itself. This can be used to give almost a converse to a well known
result of Arezzo and Pacard in the theory of constant scalar curvature
Kaehler metrics.
Andrew Strangeway (Imperial College)
A Reconstruction Theorem for the Quantum Cohomology of Fano Bundles
A vector bundle E is said to be Fano if the projectivisation P(E) is a Fano Manifold. I will present a reconstruction theorem for Fano vector bundles, which recovers the small quantum cohomology of the projectivisation of the bundle from a small number of low degree GromovWitten invariants. In special cases the quantum cohomology is entirely determined by this theorem. I will give an example where the theorem is used to calculate the quantum cohomology of a certain Fano 9fold.
Tom Sutherland (University of Oxford)
Stability conditions for the onearrow quiver. Stability
conditions are needed in order to construct nice moduli spaces, the
classical example being vector bundles over a curve. Spaces of
stability conditions of CalabiYau threefolds are also important in
studying mirror symmetry which is a duality for CalabiYau threefolds
arising in string theory. In this talk we will give an introduction
to stability conditions in algebraic geometry and then study the space
of stability conditions of a particularly simple CY3 category
described by the onearrow quiver
Affine cubic surfaces and cluster varieties In this talk we
will consider affine cubic surfaces obtained as the complement of
three lines in a cubic surface where it intersects a tritangent plane.
We will interpret certain families of these affine cubic surfaces as
moduli spaces of local systems on the punctured Riemann sphere. We
will see how to draw quivers on the sphere so that the associated
cluster variety is related to the total space of these families.
Rosemary Taylor (University of 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.
Elisa Tenni (University of Warwick)
Surface fibrations and their relative canonical
algebras. The aim of this talk is to introduce some
properties of the relative canonical algebra of a surface
fibration. It has been shown (by works of Konno, Reid, Catanese
and Pignatelli, and others) that this algebra encodes important
information about the geography of the surface. In particular I
will show how such methods apply to the case of a fibration with
genus 5 fibres, and I will prove a relation between the most
important invariants of the surface.
Alan Thompson (University of Oxford)
Tjurina and Milnor numbers of matrix singularities.
The Tjurina and Milnor numbers are two numbers that arise in the study of the singularity theory of composed
mappings. This talk aims firstly to define these numbers and provide the means to calculate them in specific
examples. This will then lead into a discussion of the fascinating relationship between the two numbers,
focussing specifically on the case where the spaces in consideration are spaces of matrices and one of the
functions to be composed is the determinant function. Here one can obtain explicit formulas relating the two
numbers in certain dimensions, but little is known about the general case.
Models for Threefolds Fibred by K3 surfaces of Degree Two.
It is well known that a K3 surface of degree two can be seen
as a double cover of the complex projective plane ramified over a smooth
sextic curve. This talk will be concerned with finding explicit
birational models for threefolds that admit fibrations by such surfaces.
It will be shown that the nature of K3 surfaces of degree two allows
these models to be constructed as double covers of rational surface
bundles, a structure which in turn enables many of their properties to
be explicitly calculated.
Andrey Trepalin (HSE, Moscow)
Rationality of the quotient of P^2 by a finite group
of automorphisms over an arbitrary field of characteristic zero
It's well known that any quotient of P^2 by a finite group is rational over an algebraically closed field. We will prove that any quotient of P^2 is rational over an arbitrary field of characteristic 0.
Jorge Vitoria (University of Warwick)
tstructures and coherent sheaves.
Let D be the derived category of coherent sheaves on a projective variety X. In this talk we will study methods of constructing tstructures on D and explore examples.
Anna Lena Winstel (TU Kaiserlautern)
The Relative Tropical Inverse Problem for Curves in a Fixed Plane.
Tropical Geometry is a rather new tool in algebraic geometry, in which an algebraic variety is assigned a polyhedral complex, called its tropical variety. By studying these tropical varieties, one can obtain information about the original algebraic variety. Since these new objects are combinatorial, problems can often be solved more easily. There is hope to find new results in algebraic geometry by looking at the tropical counterpart and then transferring the result into algebraic geometry. However, it is not always clear how this transformation from tropical into algebraic geometry can work. This problem is called the Tropical Inverse Problem: given a polyhedral complex, one asks if it is the tropical variety of an algebraic variety. So far, there are answers to this problem for special cases such as a polyhedral complex of codimension one or a polyhedral fan of dimension one, but there is no general solution. One may also ask the question in a relative setting: given an algebraic variety X and a polyhedral complex which is settheoretically contained in the tropical variety trop(X) assigned to X, does there exists a subvariety of X such that this polyhedral complex is the tropical variety of this subvariety? This question is called the Relative Tropical Inverse Problem. It is the aim of this talk to present an algorithm able to decide the Relative Tropical Inverse Problem in the case that X is the projective plane V(x+y+z+w).
John Wunderle (University of Liverpool)
Properties of higher genus curves.
We will investigate some of the geometric and number theoretic properties of curves of genus two, which admit
various types of isogenies. We will look at these via covering techniques and go on to extend some of the results
regarding curves with bad reduction at 2 and p, where p is some prime.
Jacobians of hyperelliptic curves.
The resolution of Diophantine equations over the rationals is one with a deep history. In this talk I will
consider ways to solve a general family of curves  specifically the Fermat Quartic curves ($x^4+y^4=c$). We
present work from Flynn and Wetherell and expand upon their work. We consider a "flow diagram"
approach to solving these curves and present explicit examples as well as a general method for approaching
the curves. Finally we explain how these methods can be adopted to suit other diophantine equations over the
rationals.
Christian Wuthrich (University of Cambridge)
On padic heights in families of elliptic curves.
About twenty years ago, following an initial idea of Bernardi and Neron, PerrinRiou and
Schneider found a canonical padic height pairing on an elliptic curve defined over a
number field. The associated padic regulator appears in the padic version of the conjecture
of Birch and SwinnertonDyer, but it is still unknown if this pairing is nondegenerate
except for special cases. Following the work done for the realvalued pairing, one can
analyse the behaviour of the padic height as a point varies in a family of elliptic curves,
and get so new information about this pairing.
This page is maintained by Aurelio Carlucci (carlucci at maths dot ox dot
ac dot uk).
