# V BrAG meeting

BrAG is an established series of regular meetings of British algebraic geometers. Our goal is to further strengthen the British algebraic geometry community, integrating postgraduate students and young researchers. The meetings feature a number of pre-talks for graduate students, poster sessions, career development sessions, and include plenty of time for informal interactions between the participants.

**Upcoming meeting**

The 5th BrAG meeting will take place at Imperial College in London, 14-16 September 2022. To register please fill out the form here. Deadline for full consideration for funding is 15 June 2022.

**Speakers**

Ana-Maria Castravet (Université de Versailles)

Meng Chen (Fudan University)

Alessio Corti (Imperial College)

Daniel Halpern-Leistner (Cornell University)

Frances Kirwan (Oxford University)

Maxim Kontsevich (IHES)

Dhruv Ranganathan (University of Cambridge)

Julius Ross (University of Illinois at Chicago)

Evgeny Shinder (University of Sheffield)

Balázs Szendrői (University of Vienna)

**Organisers: **Paolo Cascini (Imperial), Anne-Sophie Kaloghiros (Brunel), Cristina Manolache (Sheffield), Nicola Pagani (Liverpool),

**Scientific advisors:** Tom Bridgeland (Sheffield), Ivan Cheltsov (Edinburgh), Mark Gross (Cambridge), Miles Reid (Warwick), Richard Thomas (Imperial).

# Schedule

All the talks will be held at 180 Queen's Gate (click here for directions), in the Huxley building, room 340.

**Wednesday, 14th September**

12pm - 1pm Registration

1pm - 1:50pm Balázs Szendrői

2pm - 2:50pm Evgeny Shinder

3pm - 3:30pm Coffee Break

3:30pm - 4:20pm Julius Ross

4:45pm - 5:45pm Poster session (Huxley building - rooms 341-342)

6pm Conference Dinner (Physics common room - 819 Blackett).

**Thursday, 15th September**

9am - 9:30 am Pretalk

9:30am -10:20am Dhruv Ranganathan

10:30am - 11am Coffee Break

11am - 11:50 am Meng Chen (online)

12pm - 2pm Lunch Break

2pm - 2:50pm Alessio Corti

3pm - 3:30pm Coffee Break

3:30pm - 4:20pm Maxim Kontsevich

4:30pm - 5:00pm Pretalk

5:00pm - 5:50pm Frances Kirwan

6pm - 7pm Professional Development Session.

**Friday, 16th September**

9am - 9:30 am Pretalk

9:30am -10:20am Daniel Halpern-Leistner ( Slides )

10:30am - 11am Coffee Break

11am - 11:50 am Ana-Maria Castravet.

# Titles and Abstracts

**Ana-Maria Castravet **- Exceptional collections on moduli spaces of stable rational curves

Abstract: I will discuss joint work with Jenia Tevelev proving a conjecture of Orlov, namely that the derived category of the Grothendieck-Knudsen moduli space M(0,n) of stable, rational curves with n markings admits a full, strong, exceptional collection that is invariant under the action of the symmetric group S_n. A consequence of the conjecture is the existence of an S_n invariant basis in the cohomology of M(0,n) (in particular, the S_n representation given by the cohomology is a permutation representation).

**Meng Chen **- On minimal varieties growing from quasi-smooth weighted hypersurfaces

Abstract: We establish an effective nefness criterion for the canonical divisor of a weighted blow-up over a weighted hypersurface, from which we construct plenty of new minimal $3$-folds including $59$ families of minimal $3$-folds of general type, several infinite series of minimal $3$-folds of Kodaira dimension $2$, $2$ families of minimal $3$-folds of general type on the Noether line, and $12$ families of minimal $3$-folds of general type near the Noether line. This is a joint work with Chen Jiang and Binru Li.

**Alessio Corti **- How to make log structures

Abstract: I will make a statement about smoothing reducible toric Fano 3-folds by endowing them with certain singular log structures. I will then zoom in on a part of the proof, involving constructing (partial) resolutions of these log structures. This is joint work with Helge Ruddat.

**Daniel Halpern-Leistner **- The noncommutative minimal model program

Abstract: There are many situations in which the derived category of coherent sheaves on a smooth projective variety admits a semiorthogonal decomposition that reflects something interesting about its geometry. I will present a new unifying framework for studying these semiorthogonal decompositions using Bridgeland stability conditions. Then, I will formulate some conjectures about canonical flows on the space of stability conditions that imply several important conjectures on the structure of derived categories, such as the D-equivalence conjecture and Dubrovin's conjecture.

**Frances Kirwan** - Higher quivers and their representations

Abstract: This talk will describe ongoing work with Vidit Nanda on the concepts of higher quivers (especially 2-quivers) and their representations. The aim, if time permits, is to illustrate this using Morse theory (the original motivation for the work), geometric invariant theory and moduli of vector bundles over algebraic curves.

**Maxim Kontsevich **-** **Blow-up formula and its applications

Abstract: I'll talk about a hypothetical blow-up formula for rational Gromov-Witten invariants, give the reasons why it "should be" true, and describe its potential applications to birational geometry and to definitions of A and B models for singular varieties.

**Julius Ross **- Hodge-Riemann Bilinear Relations and Schur Classes

Abstract: The classical Hodge-Riemann bilinear relations are statements about the intersection form associated to the self-wedge product of a K\"ahler form on a compact complex manifold. Gromov initiated the question as to whether there are other cohomology that give rise these same bilinear relations, and proved that this is the case for the intersection of (possibly different) K\"ahler classes. In this talk I will discuss joint work with Matei Toma in which we generalize this to Schur classes in various ways (for instance we prove that Schur classes of ample vector bundles as well as Schur polynomials of Kahler classes all satisfy the Hodge-Riemann bilinear relations on $H^{1,1}$). This gives rise to a number of new inequalities among characteristic classes of ample vector bundles that should be thought of as generalizations of the Khovanskii-Tessier inequalities.

**Evgeny Shinder **- Categorical ordinary double points

Abstract: I will explain how to construct semiorthogonal decompositions of the derived category of a nodal variety into a smooth proper category and several copies of the categorical ordinary double points, as well as numerical obstructions to such decompositions.This is joint work with A. Kuznetsov.

**Dhruv Ranganathan** - The cycle of curves in a toric variety

Abstract: Natural cycles in the moduli space of curves arise from examining the loci where curves admit a map to a fixed variety X. I will discuss the cycle of curves that is obtained when X is toric. When X is P1, the story is now well-understood as the double ramification cycle, and the focus here is on what happens when X has dimension 2 or larger. The study of these issues leads naturally to the development of logarithmic intersection theory. I will discuss the outcome of this development, including connections to enumerative geometry, such as Hurwitz numbers and Severi degrees, to the tautological ring of the moduli space of curves, and to tropical correspondence theorems. During the pre-talk I will give an introduction to the double ramification cycle via Gromov-Witten theory.

**Balázs Szendrői**** **- ADE singularities, Quot schemes and generating functions

Abstract: Starting with an ADE singularity C^2/Gamma for Gamma a finite subgroup of SL(2,C), one can build various moduli spaces of geometric and representation-theoretic interest as Nakajima quiver varieties. These spaces depend in particular on a stability parameter; quiver varieties at both generic and non-generic stability are of geometric interest. Generating functions of Euler characteristics at different points in stability space are controlled by specialisation formulae of characters of Lie algebra representations. We will explain some of these connections, focusing in particular on the abelian case. Based on joint papers and projects with Craw, Gammelgaard, Gyenge, and Nemethi.