# Algebraic Geometry Seminar

The algebraic geometry seminar in Term 3 2023/2024 will usually meet on Wednesdays at 3pm, though we may sometimes change to allow speakers from other time zones.

**Each talk will be either in B3.02 or MB0.07 -- see specific information below.**

A link to talks from the previous term is here.

**Wednesday 24 April 2024 3pm, B3.02.**

**Speaker: **Barbara Fantechi (SISSA)

**Title: **Deformations of semi-smooth varieties and the boundary of the moduli space of Godeaux surfaces

**Abstract: **A variety X is semismooth if étale locally it is isomorphic to a product of a pinch point (x^2y-z^2) with some affine space; equivalently, its normalization is smooth and X is obtained by gluing a smooth divisor to itself via an involution with fixed points in codimension 1. In joint work with Marco Franciosi and Rita Pardini, we calculate the sheaves T^1_X and T_X in terms of the normalization and the gluing, and use this to show that all semi-smooth non normal stable Godeaux surfaces are smoothable, and nonsingular points of the moduli space.

**Wednesday 1 May 2024 4pm (Note unusual time)**

**Speaker: **Lucie Devey (Edinburgh)

**Title: **Toric vector bundles, Parliament of polytopes and Stability

**Abstract: **Given any toric vector bundle, we may construct its parliament of polytopes. This is a generalization of the moment polytope of a toric line bundle. It contains a huge amount of information about the original bundle: notably on its global sections and its positivity. In this talk, we explain how to determine algorithmically if a fixed toric vector bundle is semi-stable or not (with respect to any polarisation), we illustrate it on its parliament of polytopes. This is a first step in getting a classification of toric vector bundles.

**Wednesday 8 May 2024 3pm.**

** Speaker: -**

**Title:** -

**Wednesday 15 May 2024 3pm, B3.02. (postponed due to the illness of the speaker)**

**Wednesday 22 May 2024 3pm. **

**Speaker: -**

**Title**: -

**Wednesday 29 May 2024 3pm. **

**Speaker: Patrick Kennedy-Hunt** (Sheffield)

**Title:** Degenerating Hilbert schemes and the secondary fan**Abstract:** I will start by explaining the secondary fan of a polytope and its connection to a moduli space of tropical curves. Associated to any fan is a toric variety, and I will explain how to interpret the fan associated to a moduli space of tropical curves as a natural moduli space. Time permitting, I will outline how this toric story is a special

case of a theory built to study Hilbert schemes through degeneration techniques.

**Wednesday 5 June 2024 3pm.**

** Speaker: -**

**Title:** -

**Wednesday 12 June 2024 3pm. **

**Speaker: Andrés Ibáñez Núñez **(Oxford)

**Title:**Refined Harder-Narasimhan filtrations in moduli theory

**Abstract:**We introduce a notion of refined Harder-Narasimhan filtration, defined abstractly for algebraic stacks satisfying natural conditions. Examples include moduli stacks of objects at the heart of a Bridgeland stability condition, moduli stacks of K-semistable Fano varieties, moduli of principal bundles on a curve, and quotient stacks. We will explain how refined Harder-Narasimhan filtrations are closely related both to stratifications and to the asymptotics of certain analytic flows, relating and expanding work of Kirwan and Haiden-Katzarkov-Kontsevich-Pandit, respectively. In the case of quotient stacks by the action of a torus, the refined Harder-Narasimhan filtration can be computed in terms of convex geometry.

**Tuesday 18 June 2024 3.15pm. (B1.01)**

**Speaker: Hexu Liu **(Fudan)

**Title**: A lifting principle for canonical stability indices of varieties of general types

**Abstract:**

For any positive integer n, the n-th canonical stability index r_n is defined to be the smallest number such that the r_n-canonical map is stably birational onto its image for all smooth projective n-folds of general type. In this talk, I will introduce a "lifting principle" for projective varieties of general type: r_n equals to the maximum of the set of those canonical stability indices of smooth projective (n+1)-folds with sufficiently large canonical volumes. Equivalently, there exists a constant K(n) > 0 such that, for any smooth projective n-fold X with the canonical volume vol(X) > K(n), the pluricanonical map phi_{m,X} is birational onto the image for all m >= r_{n-1}.

**Tuesday 18 June 2024 4.45pm. (B1.01)**

**Speaker: Stephen Coughlan **(Mary Immaculate College, Limerick)

**Title**: Threefolds on the Noether line

**Abstract: **

The 3-dimensional analogue of Noether's inequality for surfaces has recently been proven by Chen, Chen, Jiang. The inequality is sharp, and those threefolds which satisfy the equality are said to lie on the Noether line. I report on a project to describe and classify all threefolds on the Noether line, and possibly all threefolds close to the

line as well. This is joint work with Roberto Pignatelli, Yong Hu and Tong Zhang.

**Wednesday 19 June 2024 3pm. (B3.02)**

**Speaker: Mura Yakerson** (Oxford)

**Title: **Motivic Adams conjecture

**Abstract:**

The well-known Adams conjecture in topology is a theorem about compactifications of real vector bundles on CW-complexes, which has important implications for analyzing stable homotopy groups of spheres. In the talk we will discuss an algebro-geometric version of this statement, which tackles algebraic vector bundles on smooth algebraic varieties. This is joint work with Alexey Ananyevskiy, Elden Elmanto and Oliver Röndigs.

**Wednesday 26 June 2024 3pm. **

**Speaker: Simon Pepin Lehalleur** (Amsterdam)

**Title:** Motivic mirror symmetry for Higgs bundles in arbitrary characteristic**Abstract:**

Higgs bundles are vector bundles equipped with an additional "twisted endomorphism". Moduli spaces of Higgs bundles on a smooth projective curve have a very rich geometry that is both related to the geometry of moduli of vector bundles but also has additional symplectic features. I will introduce these moduli spaces and discuss some of what is known about their cohomology and their motivic invariants. There has been a lot of recent progress in this direction and I will try to describe the main threads. I will conclude with a discussion of my joint work with Victoria Hoskins on a motivic version of the "cohomological mirror symmetry" conjecture of Hausel and Thaddeus for SL_n and PGL_n Higgs bundles, including in positive characteristic.