Warwick Algebraic Geometry Seminar

Abstracts

Qingyuan Jiang (Chinese University of Hong Kong) - Categorical Plucker Formula and Homological Projective Duality

I will talk about a recent joint work with Prof. Conan Leung and Mr. Ying Xie.

In the classical theory of topology of a projective algebraic variety X, on one hand, we have Lefschetz theory for linear sections, which states the cohomology of a hyperplane section almost all comes from the cohomology of the ambient space X except for middle dimension, where new interesting contributions emerge. On the other hand, we have Plucker formula for non-linear sections, which states the Euler characteristics of the transverse intersection X ∩ T of X by another subvariety T can be expressed in terms of the ones of the intersection of their projective dual varieties X^v ∩ T^v.

For the theory of bounded derived category of coherent sheaves, on one hand, Kuznetsov introduced the concept of Homological Projective Duality (HPD) to study linear sections, and showed the HPD-theorem, which states when the HPD of X exists as a variety Y, not only we have Lefschetz type decompositions for the derived categories of linear sections X_L⊥, but also descriptions of interesting parts in terms of derived categories of the dual linear sections Y_L of Y.

For non-linear sections, we show that HPD theorem still holds for non-linear sections of the pair (X,Y) by another HP-dual pair (S,T), as long as if they intersect properly. This enables us to produce examples of decomposition of derived category of varieties as well as give descriptions of their interesting parts. When taking Euler characteristic of Hochschild homology of our formal, we obtain Plucker type formula for HPD. If we take (S,T) to be dual linear projective subspaces, our method gives a more direct and categorical proof of Kuznetsov's original HPD theorem without referring concrete geometries of Grassmannians.

Miles Reid (University of Warwick) - Proof of Riemann-Roch for Curves + Organisational Meeting

I will discuss the proof of the Riemann-Roch Theorem for curves. This will be followed by a short organisational meeting where we will discuss the rest of the term's seminars.

Christian Boehning (University of Warwick) - Conic Bundles over Rational Threefolds with Nontrivial Unramified Invariants

We discuss a construction scheme for conic bundles over P³ that have nontrivial unramified Brauer group and mildly singular total spaces (meaning, admitting a Chow zero universally trivial resolution). This gives new families of rationally connected fourfolds, the very general member of which is not stably rational. We also explain a recent example by Hassett-Tschinkel-Pirutka from this new perspective. The latter example was used to prove that rationality is not deformation-invariant in families of smooth varieties. This had been famously open for a long time.

Gregory Sankaran (University of Bath) - Type III Degenerations of Hyperkähler Manifolds

Degenerations of K3 surfaces were studied by Kulikov and by Pinkham and Persson in the 1970s: there are two distinct genres, called Type II and Type III. A similar phenomenon occurs for irreducible hyperkähler manifolds generally. I shall describe some preliminary results about the Type III case, which has been less studied, and some possible consequences.

Gavin Brown (University of Warwick) - Introduction to Unprojection

This will be an introductory talk on the topic of unprojection, aimed at graduate students.

Gavin Brown (University of Warwick) - Fanos in P² x P² Format

Fano varieties can be embedded in weighted projective space 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) disciples of Miles Reid’s graded rings program have found a few hundred deformation families of Fano 3-folds to realise all the Hilbert series in those cases. Sometimes more than one deformation family may realise a given Hilbert series. I find some families of Fano 3-folds whose equations look like those of the Segre embedding of P² x P² (so lie in codimension 4) that seem to be new. (This is in progress, and joint with Al Kasprzyk and Imran Qureshi.)

Miles Reid (University of Warwick) - More Graded Rings

This talk will, in some sense, be a follow on from my talk at the "3 C in G" event from Friday 14th October. It will discuss graded ring constructions, and will be aimed at graduate students.

Nicholas Shepherd-Barron (King's College London) - Henon Maps and Feistel Ciphers: Cremona Transformations in Cryptography

The point of this talk is to point out that certain elementary Cremona transformations are the building blocks of encryption algorithms, both recent (DES) and current (AES), and to suggest, and ask for, ways in which algebraic geometry can be used to analyze their security.

(This is *not* a talk on public key cryptography.)

Alan Thompson (University of Warwick) - Introduction to K3 Surfaces

I will present an introduction to K3 surfaces. This will probably start with some basic geometry, then move on to discuss (lattice) polarizations and/or moduli spaces. There will be lots of examples. This talk will be aimed at graduate students.

Brent Pym (University of Oxford) - Poisson Structures on Fano Manifolds

A Poisson variety is an algebraic variety equipped with a Poisson bracket on its regular functions. Such a variety carries a natural foliation by symplectic submanifolds. For projective spaces and other Fano manifolds, this foliation is typically highly singular. For example, a conjecture of Bondal predicts that the dimensions of the singular strata are much greater than one would expect from the classical theory of degeneracy loci of bundle maps. I will describe some progress on this conjecture, and related results concerning the classification of low-dimensional Poisson varieties, where elliptic curves feature prominently.

Michele Nicolussi (Universität Tübingen) - Terminal Fano Threefolds of Complexity One

We aim to classify the terminal Q-factorial Fano threefolds that come with an effective action of a two-dimensional torus. Generalizing the correspondence between toric Fano varieties and lattice polytopes, we associate to any Fano variety of complexity one a certain polyhedral complex. The lattice points inside it control the discrepancies. This leads to a simple characterization of terminality and canonicity. We report the latest results in the classification: threefolds with Picard number one and the larger family of threefolds that do not admit any birational divisorial contraction.

Jinhyung Park (KIAS) - Newton-Okounkov Bodies and Asymptotic Invariants of Divisors

A Newton-Okounkov body is a convex body in Euclidean space associated to a divisor on an algebraic variety with respect to an admissible flag. After briefly recalling basics of Newton-Okounkov bodies of ample or big divisors, I introduce two natural ways to associate Newton-Okounkov bodies to pseudoeffective divisors. We then study various asymptotic invariants of pseudoeffective divisors using these convex bodies. This is joint work with Sung Rak Choi, Yoonsuk Hyun, and Joonyeong Won.

Nicola Pagani (University of Liverpool) - Wall-Crossing on Universal Compactified Jacobians

The last 20 years have seen huge developments in the enumerative geometry of the moduli spaces M_g,n of stable curves. In this talk, we will discuss the beginning of a similar programme for the universal Jacobian J_g,n parameterizing line bundles on stable curves. The universal Jacobian admits many natural compactifications, each of which should play an important role in the enumerative geometry, thus giving rise to interesting wall-crossing phenomena. We will discuss our first results in this research programme and their application. We have an explicit picture of the combinatorics of the stability space and of the walls that govern all different compactifications, and we understand how the wall-crossing works for codimension-1 cycles. This is research (partly in progress) with Jesse Kass (South Carolina).

Joe Karmazyn (University of Sheffield) - Simultaneous Resolutions and Noncommutative Algebras

Minimal resolutions of surface quotient singularities can be studied and understood via noncommutative algebra in a variety of ways packaged as the McKay correspondence. Assorted higher dimensional examples, such as flopping contractions of 3-folds, can be realised from simultaneous resolutions associated to surface quotient singularities. I will discuss how these simultaneous resolutions can also be understood via noncommutative algebras, and how certain examples can be easily calculated.

Elisa Postinghel (Loughborough University) - Tropical Compactifications, Mori Dream Spaces and Minkowski Bases

Joint work in progress with Stefano Urbinati.

Given a Mori Dream Space X, we construct via tropicalisation a model dominating all the small QQ-factorial modifications of X. Via this construction we recover a Minkowski basis for the Newton-Okounkov bodies of Cartier divisors on X and hence generators of the movable cone of X.

Robert Marsh (University of Leeds) - Twists of Plücker Coordinates as Dimer Partition Functions

Joint work with Jeanne Scott.

The homogeneous coordinate ring of the Grassmannian Gr(k,n) has a cluster algebra structure defined in terms of certain planar diagrams known as Postnikov diagrams. The cluster associated to such a diagram consists entirely of Plücker coordinates.

We introduce a twist map on Gr(k,n), related to the twist of Berenstein-Fomin-Zelevinsky, and give an explicit Laurent expansion for the twist of an arbitrary Plücker coordinate in terms of a scaled matching polynomial. This matching polynomial arises from the bipartite graph dual to the Postnikov diagram of the initial cluster, modified by an appropriate boundary condition.

Thomas Prince (Imperial College London) - A Symplectic Approach to Polytope Mutation

Polytope mutation is a combinatorial operation which appeared in the study of the birational geometry of Landau-Ginzburg models 'mirror-dual' to Fano manifolds. We give a mirror/symplectic account of this subject. This heavily utilizes the notion of an almost-toric Lagrangian fibration, due to Symington. This perspective elucidates the connection with cluster and quiver mutation (in the surface case) as well as the connection to toric degenerations via 'algebraization' techniques due to Gross-Siebert.

Getting Here

Directions to the university may be found here. Once you're on campus, the Mathematics Institute is located in the Zeeman building; you can download a map of the campus here.

Please note that if you are arriving by public transport, the University of Warwick is not in fact in the town of Warwick, or indeed anywhere near it. Instead, it is located a short distance southwest of Coventry. If you are coming by train the closest stations are Coventry and Leamington Spa.

To get to campus from Coventry station you should take bus 11, 11U, or 12X; all three leave from stand ER3 at the bus hub outside the railway station. At the time of writing, a single ticket from Coventry station to the university costs £2; please note that the buses from Coventry only accept exact change.

To get to campus from Leamington Spa station you should take bus U1, U2, or U17. Please note that these buses do not leave from directly outside the station; instead, the nearest bus stop is just around the corner on Victoria Terrace. A map of the route may be found here. At the time of writing, a single ticket from Leamington Spa station to the university costs £2.75.