*Harvard**–**MIT Algebraic Geometry Seminar*

*Spring 2021*

*Tuesdays at 3 pm*

*Schedule: (Tuesday 3-4 pm unless otherwise noted*)*

*Schedule: (Tuesday 3-4 pm unless otherwise noted*)*

Please signup to our mailing list here if you would like to attend.

*Feb 9,** ***Zoom***: *Gavril Farkas, *Humboldt-Universität zu Berlin. ***The Kodaira dimension of the moduli space of curves: recent**** ****progress on a century-old problem.**

### Abstract:

The problem of determining the birational nature of the moduli space of curves of genus g has received constant attention in the last century and inspired a lot of development in moduli theory. I will discuss progress achieved in the last 12 months. On the one hand, making essential use of tropical methods it has been showed that both moduli spaces of curves of genus 22 and 23 are of general type (joint with D. Jensen and S. Payne). On the other hand I will discuss a proof (joint with A. Verra) of the uniruledness of the moduli space of curves of genus 16.

*Feb 16,** ***Zoom***: *Shamil Asgarli, *Univ. of British Columbia.* **On the proportion of transverse-free curves.**

### Abstract:

Given a smooth plane curve C defined over an arbitrary field k, we say that C is transverse-free if it has no transverse lines defined over k. If k is an infinite field, then Bertini's theorem guarantees the existence of a transverse line defined over k, and so the transverse-free condition is interesting only in the case when k is a finite field F_q. After fixing a finite field F_q, we can ask the following question: For each degree d, what is the fraction of degree d transverse-free curves among all the degree d curves? In this talk, we will investigate an asymptotic answer to the question as d tends to infinity. This is joint work with Brian Freidin.

*Feb 23 @ 4:30pm*,** ***Zoom***: *Xiaolei Zhao, *UC Santa Barbara. ***Elliptic quintics on cubic fourfolds, moduli spaces of O'Grady 10 type, and intermediate Jacobian fiberation.**

### Abstract:

In this talk we study certain moduli spaces of semistable objects in the Kuznetsov component of a cubic fourfold. We show that they admit a symplectic resolution \tilde{M} which is a smooth projective hyperkaehler manifold deformation equivalent to the 10-dimensional example constructed by O’Grady. As a first application, we construct a birational model of \tilde{M} which is a compactification of the twisted intermediate Jacobian fiberation of the cubic fourfold. Secondly, we show that \tilde{M} is the MRC quotient of the main component of the Hilbert scheme of elliptic quintic curves in the cubic fourfold, as conjectured by Castravet. This is a joint work with Chunyi Li and Laura Pertusi.

*Mar 2,** ***Zoom***: *Ruijie Yang*, Stony Brook University. ***Decomposition theorem for semisimple local systems.**

### Abstract:

In complex algebraic geometry, the decomposition theorem asserts that semisimple geometric objects remain semisimple after taking direct images under proper algebraic maps. This was conjectured by Kashiwara and is proved by Mochizuki and Sabbah in a series of long papers via harmonic analysis and D-modules. In this talk, I would like to explain a more geometric/topological approach in the case of semisimple local systems adapting de Cataldo-Migliorini. As a byproduct, we can recover a weak form of Saito's decomposition theorem for variations of Hodge structures. Joint work in progress with Chuanhao Wei.

*Mar 9*, * Zoom*: Mihai Paun,

*Univ. of Bayreuth.*

**On extension of pluricanonical forms for Kaehler families.**

### Abstract:

We will report on a recent joint work with Junyan Cao, cf. arXiv:2012.05063. The main topics we will discuss are revolving around the extension of pluricanonical forms defined on the central fiber of a family of Kaehler manifolds. For our results to hold we need the divisor of zeros of the said forms to be suficiently "nice", in a sense that will become clear during the talk.

*Mar 16 @ 4:30pm*,** ***Zoom***: *Lukas Brantner, *Univ. of Oxford. ***Lie algebras, deformations, and Galois theory in characteristic p.**

### Abstract:

We introduce a derived version of Lie algebras in characteric p and describe two recent applications: first, we use them to classify infinitesimal deformations, generalising the Lurie-Pridham theorem in characteristic zero; second, we prove a Galois correspondence for purely inseparable field extension, extending work of Jacobson at height one. This talk is based on joint works with Mathew and Waldron.

*Mar 23,** ***Zoom***: *Chiara Damiolini, *Rutgers University. ***Conformal blocks from vertex algebras of CohFT-type.**

### Abstract:

In this talk I will discuss properties of certain vector bundles on the moduli space of stable n-pointed curves which arise from admissible modules over certain vertex operator algebras. I will in particular describe conditions on the vertex algebra that guarantee that the factorization property holds for these vector bundles and discuss its consequences and open problems if time permits. This is based on a joint work with A. Gibney and N. Tarasca.

*Mar 30,** ***Zoom***: *Mihnea Popa,* Harvard. ***Hodge filtration on local cohomology and applications.**

### Abstract:

This describes joint work in progress with M. Mustata, in which we study the filtration on local cohomology sheaves induced by their natural mixed Hodge module structure. Special properties of this filtration, for instance strictness, lead to a number of different applications, including an injectivity theorem for dualizing complexes, local vanishing for forms with log poles, and especially a characterization of the local cohomological dimension of a closed subscheme in terms of data arising from a log resolution of singularities.

*Apr 6,** ***Zoom***: *Charlotte Chan, *MIT. ***Flag varieties and representations of p-adic groups.**

### Abstract:

Algebraic geometry has had a huge influence on representation theory for (nearly) the last century. I'll talk about some historical instances of this relationship and recent progress in translating our algebraic understanding of representations of p-adic groups to geometric contexts. Parts of this talk are based on joint work with A. Ivanov and joint work with M. Oi.

*Apr 13,** ***Zoom***: *Soheyla Feyzbakhsh, *Imperial College London. ***Application of a Bogomolov-Gieseker type inequality to counting invariants.**

### Abstract:

I will work on a Calabi-Yau 3-fold X which satisfies the Bogomolov-Gieseker conjecture of Bayer-Macr\`i-Toda for weak stability conditions, such as the quintic threefold. I will explain how wall-crossing with respect to weak stability conditions gives an expression of Joyce’s generalised Donaldson-Thomas invariants counting Gieseker semistable sheaves of any rank greater than or equal to one on X in terms of those counting sheaves of rank 0 and pure dimension 2. This is joint work with Richard Thomas.

*Apr 20,** ***Zoom***: *

### Abstract:

*Apr 27,** ***Zoom***: *Ziquan Zhuang, *MIT.*

### Abstract:

*May 4,** ***Zoom***: *Stefan Schreieder, *University of Hannover. *