# JAWS Past Talks 2018/19

## 2018 Term 1

Week 1, 05/10/18

### Moduli spaces of stable pairs on the conifold and quiver representations

A quiver is a collection of arrows, that is an oriented graph. They appear in different branches of mathematics and theoretical physics, as a means to encode, in an efficient way, combinatorial information arising from algebraic or categorical data. In certain cases, they allow to describe moduli spaces appearing in algebraic geometry.

This talk is going to be an introduction on the latter aspect, with a perspective on my current research. We will start by considering classical examples, like the Kronecker quiver or the trefle quiver, and the corresponding moduli spaces, hinting at the applications to enumerative geometry.

We will move then to the topic of representations of the path algebra associated to a quiver, and the notion of stability introduced by King; there is indeed a remarkable translation, in the quiver case, of the more general notion of stability for an abelian category.

In particular, Nagao and Nakajima proved that some classes of moduli spaces of sheaves on the resolved conifold, appearing in enumerative geometry, can be interpreted as moduli spaces of representations of a certain framed quiver, allusively called conifold quiver. We will unravel the details of this association, showing how simple computations can lead to the characterisation of stability conditions for representations of the conifold quiver.

This leads to an explicit coordinate description of the moduli spaces of stable pairs on the resolved conifold: we confirm the results for the known cases, and we complement the information already obtained by a more geometric approach in a new case.

Week 2, 8/10/2018

### Generalised braid categorification

Ordinary braid group Brn is a well-known algebraic structure which encodes configurations of n non-touching strands (“braids”) up to continuous transformations (“isotopies”). A classical result of Khovanov and Thomas states that this group acts categorically on the space Fln of complete flags in Cn. Generalised braids are the braids whose strands are allowed to touch in a certain way. They have multiple endpoint configurations and can be non-invertible, thus forming a category rather than a group. In this talk I will present some progress that have been made towards extending the result of Khovanov and Thomas to the categorification of the generalised braid category.

Week 3, 15/10/2018

### Symmetric powers of varieties and rational equivalence of 0-cycles

For any quasi-projective variety X over a field k, its d-th symmetric power is the quotient of the its d-th Cartesian product by the action of the permutation group of d elements.

In particular, if the field is algebraically closed, the k-points of the d-th symmetric powers of the variety correspond to a zero-cycle of degree d on X. Moreover, rational equivalence of zero cycles is encoded by rational curves on symmetric powers of X.

In this talk, we review a classical result by Mumford stating that if X is a nonsingular projective surface with positive geometric genus, its group of 0-cycles of degree 0 modulo rational equivalence is not finite dimensional.

Week 4, 22/10/2018

### Mirror Symmetry for Homogeneous Spaces

In this talk we will take a look at mirror symmetry at the level of the small quantum cohomology, specialized to the case of homogeneous spaces. We will introduce Gromov-Witten invariants and how they are involved in the small quantum cohomology using complex projective space as an illustration. We will stay with projective space, but now consider it as a homogeneous space, and work out a method to construct a mirror pair for it: that is, a variety whose coordinate ring modulo relations defined by a rational function models the small quantum cohomology of projective space. Finally, if time allows, we will try to give some insights how the construction works for more general homogeneous spaces.

Week 5, 29/10/2018

### Degenerations of Conic Bundles in any Characteristic

This talk aims to introduce informally how we can employ the recent degeneration method by Voisin to a wide class of rationality problems. After discussing the general method and its difficulties, the talk will focus on applications to conic bundles, with special regard to mixed characteristic degenerations. The special case of char = 2 will be highlighted as it was result of original research together with A. Auel, Ch. Boehning, H.Ch. Graf von Bothmer.

Week 7, 12/11/2018

### Deformation Theory

Deformation theory can be thought of as the study of infinitesimal thickenings of geometric or algebraic objects. Equivalently, it is the study of infinitesimal neighbourhoods in moduli spaces. In this talk, I'll provide an introduction to some deformation-theoretic concepts. I'll explain Deligne's philosophy that in characteristic zero, deformation problems are 'controlled' by differential graded Lie algebras (dglas). I'll talk about derived deformation theory, and how it clarifies some concepts within the classical theory, in particular the proofs of Lurie and Pridham that dglas are equivalent to derived moduli problems via Koszul duality. Time permitting, I'll talk about an application of noncommutative deformation theory to birational geometry in the form of the Donovan-Wemyss contraction algebra, and I will mention how derived noncommutative deformations also fit into the picture.

Week 8, 19/11/2018

### 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 effcient 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.

Week 9, 26/11/2018

### Multipoint Seshadri constants and explicit Kähler packings of projective complex manifolds.

In this talk I will start by introducing the concept of multipoint Seshadri constants and discuss their relationship withNagata's conjecture. In the remaining time I will introduce the notions of Symplectic and Kähler packings and end by showing that there is a direct connection between the multipoint Seshardi constant and the existence of Kähler packings.

Week 10, 3/10/2018

### James Plowman (University of Warwick)The Chow-Witt group of a scheme with dualising complex

Chow Witt groups were originally introduced by Barge & Morel as a cohomological object which they conjectured would serve as a good target for a theory of Euler classes - these classes have since been constructed in many cases of interest. In this talk we aim to define the Chow-Witt group for possibly singular schemes similarly to the description of the ordinary Chow group as cycles modulo rational equivalence.

## 2019 Term 2

Week 2, 14/01/19

### Angelica Simonetti (UMass Amherst)On cusps singularities and their smoothability

Given an isolated surface singularity it is natural to study its deformation theory and look for conditions for it to be smoothable: it turns out that for normal surface cusp singularities there's a nice result initially conjectured by Looijenga relating the existence of smoothings of such singularities to the combinatorics of certain special rational surfaces. After a general introduction on cusp singularities I will discuss Looijenga's theorem, giving a sketch of its proof.

Week 3, 21/01/19

### Birational Rigidity of higher dimensional Fano spaces

In this talk we will review some recent results in the area of Birational Rigidity. We will begin by recalling the main definitions and theorems behind the theory, before studying the case of a cyclic cover with a singular point, showing the Birational Superrigidity of a general such variety. We will also review some more recent results using the techniques discussed.

Week 5, 04/02/19

### Tim Grange (Loughborough University)Blow-ups of products of projective spaces

The classification of Mori Dream Spaces is an open problem in algebraic geometry. Castravet and Tevelev gave a criterion to determine whether the blow-up of products of projective spaces of the form (ℙ^a)^b at points in general position are Mori Dream Spaces. The case of arbitrary products of projective spaces remains unknown. In this talk, I will discuss progress in this direction and give some positivity results.

Week 6, 11/02/2019

### Newton-Okounkov bodies of exceptional curve valuations

Newton-Okounkov bodies are convex bodies associated to line bundles on projective varieties that capture positivity properties of the line bundle in question. Let $p$ be a closed point in ${\mathbb{P}}_{\mathbb{C}}^^2$ and consider a surface obtained by a sequence of finitely many blowups of points where we start with $p$ and always blow up a point in the exceptional divisor created last.
Our result is an explicit description of the Newton-Okounkov body of the pullback of $\mathcal{O}_{\mathbb{P}^^2}(1)$ with respect to the flag given by the last exceptional divisor and a point on it.

Week 7, 18/02/2019

## Hyperelliptic integrals and mirrors of the Johnson-Kollár del Pezzo surfaces

In this talk I will consider the regularised I-function of the family of del Pezzo surfaces of degree 8k+4 in P(2,2k+1,2k+1,4k+1), first constructed by Johnson and Kollár, and I will ask the following two equivalent questions:
1) Is this function a period of a pencil of curves?
2) Does the family admit a Landau-Ginzburg (LG) mirror?
After some background on the Fano-LG correspondence, I will explain why these two questions are interesting on their own, and I will give a positive answer to them by explicitly constructing a pencil of hyperelliptic curves of genus 3k+1 as a LG mirror.
To conclude, I will sketch how to find this pencil starting from the work of Beukers, Cohen and Mellit on hypergeometric functions.
This is a joint work with Alessio Corti.

Week 8, 25/02/2019

### Torus actions and Fanos

Toric varieties are well understood and classifications of Toric Fanos has been done in 2 and 3 dimensions. We extend some of these results to the case where the torus is a smaller dimension than the variety. We will discuss both how this is useful in the study of toric and non toric cases.

Week 9, 4/03/2019

## On the monodromy group of projections

Let X be a irreducible, reduced, projective variety over the complex numbers; consider the linear projection of X from a suitable linear subspace L; it is a finite map of degree d=deg(X). To it we can associate the monodromy group, that is a transitive subgroup of the symmetric group S_d.
It is known that for a general L, the monodromy group associated to the projection is S_d. We say that L with monodromy group S_d is uniform.
I will recall some results on the dimension of the locus of the non uniform centers of projections: a work on curves of Pirola and Schlesinger and a work in progress on hypersurfaces in collaboration with A. Cuzzucoli and R. Moschetti.

Week 10, 11/03/2019

## An Enriques Theorem in Characteristic p

The problem treated in this talk will be the search for numerical invariants that birationally characterize abelian varieties. Enriques proved at the beginning of the XX century that a smooth complex surface with P_1=P_4=1 and irregularity equal to 2 is birationally equivalent to an abelian surface. I will outline the history of how Enriques’ theorem has been improved and extended to higher dimensional varieties in characteristic zero and what progress has been made in characteristic p. In particular, I will discuss a first result I obtained as part of my PhD project: a version of Enriques’ theorem for surfaces in characteristic p.

Week 2, 29/04/19

## Bond theory for the analysis of mobile linkages

We introduce and explore the theory of bonds using the language of quaternions. Bond theory is a recent algebraic tool that has been successfully employed in the study of over-constrained linkages. These are mobile mechanisms that consist of rigid bodies assembled together by joints and that are subject to "too many" restrictions in its movements. From the point of view of bond theory, one aims to deduce as much algebraic information as possible so as to answer to questions like: "How can I assemble these sets of joints and rigid bodies in order to make a mobile mechanism?"
Week 4, 13/05/2019

### Stringy Invariants, Toric Varieties, and Lattice Polytopes

We present topological invariants in the singular setting for projective Q-Gorenstein varieties with at worst log-terminal singularities, such as
stringy Euler numbers, stringy Chern classes, stringy Hodge numbers, and stringy E-functions. In the toric setting, we give formulae to efficiently
compute these stringy invariants. Using these combinatorial expressions and the stringy Libgober-Wood identity, we derive several appealing new
combinatorial identities for lattice polytopes. We go on to generalise the famous ‘number 12’ and ‘number 24’ identities which hold far more
generally than previously expected.

Week 5, 20/05/2019

## Perverse Schobers

The group of autoequivalences of a given derived category of coherent sheaves on a variety X, D^b(X), has been the subject of much study. In this talk I will present some well-known results about Aut(D^b(X)), as well as some more recent technology in the form of conjectural objects known as perverse schobers. These should be thought of as categorifications of perverse sheaves, a notion which can be made more precise through results of Bondal, Kapranov and Schechtman.

Week 6, 31/05/2019

### Cohomology of the moduli space of non-hyperelliptic genus four curves

In this talk I will present the intersection Betti numbers of the moduli space of non-hyperelliptic Petri-general genus four curves. This space has a canonical compactification as GIT quotient, which was proved to be the final step in the Hassett-Keel log MMP for stable genus four curves. The strategy of the cohomological computation relies on a general method developed by F. Kirwan to calculate the cohomology of GIT quotients of projective varieties, based on stratifications, a partial desingularisation and the decomposition theorem.

Week 7, 3/06/2019

## Higher Categories in Mirror Symmetry

I will explain how $\infty$-categories with their 'Feynman' diagrams lie at the very deep foundations of String Theory and Mirror Symmetry, motivating the rising field of (noncommutative) Derived Algebraic Geometry.