# Algebraic Geometry Seminar 22/23 Term 2

The algebraic geometry seminar in Term 2 2022/2023 will usually meet on Wednesdays at 3pm in B3.02, though we may sometimes change to allow speakers from other time zones.

See the talks from the previous term here.

**Wednesday 11 January 2023, 3pm. Speaker: Jenia Tevelev (U. Massachusetts -- Amherst)**

Title: Semi-orthogonal decompositions of moduli spaces

Abstract: Let C be a smooth projective curve of genus g at least 2 and let N be the moduli space of stable rank 2 vector bundles on C with fixed odd determinant. It is a smooth Fano variety of dimension 3g-3, Picard number 1 and index 2. We construct a semi-orthogonal decomposition of the bounded derived category of N conjectured by Narasimhan and by Belmans, Galkin and Mukhopadhyay. It has two blocks for each i-th symmetric power of C for i = 0,...,g−2 and one block for the (g − 1)-st symmetric power. Our proof is based on an analysis of wall-crossing between moduli spaces of stable pairs, combining classical vector bundles techniques with the method of windows. Joint work with Sebastian Torres.

**Wednesday 18 January 2023, 3pm. Speaker: Qaasim Shafi (Birmingham)**

Title: Divisors on Logarithmic Mapping Spaces

Abstract: The stable maps compactification of the space of rational, degree d curves in P^r is used to define Gromov-Witten invariants and is helpful for solving enumerative problems concerning rational curves in projective space. Its geometry is well studied. In particular, its Picard/Class group (over Q) was determined by Pandharipande.

The space of rational, degree d curves in P^r with fixed tangencies to a hyperplane H has a compactification by stable logarithmic maps and is used to define relative (or logarithmic) Gromov-Witten invariants. In joint work with Patrick Kennedy-Hunt, Navid Nabijou and Wanlong Zheng we determine its Class group and Picard group (over Q). I will highlight some of the differences which arise in the logarithmic case and explain how this is part of a broader programme aimed at understanding the geometry of logarithmic mapping spaces.

**Wednesday 25 January 2023, 3pm. Speaker: Roberto Gualdi (Regensburg)**

Title: How complicated are the solutions of a system of polynomial equations?

Abstract: A beautiful result due to Bernstein and Kushnirenko allows to predict the number of solutions of a system of Laurent polynomial equations from the combinatorial properties of the defining Laurent polynomials.

In a joint work with Martín Sombra (ICREA and Universitat de Barcelona), we give intuitions for an arithmetic version of such a theorem. In particular, in the easy case of the planar curve x + y + 1 = 0, we show how to guess the arithmetic complexity of the intersection point between this line and its translates by torsion points.

The talk will involve a bit of height theory, special values of the Riemann zeta function and will not forget to pay homage to a piece of British literature.

~~Wednesday 1 February 2023, 3pm~~. No seminar this week

**Wednesday 8 February 2023, 3pm. Speaker: Inder Kaur (Loughborough)**

Title: Specialization of dominant maps

Abstract: In the last few years many results on the specialization of rationalilty have been shown. In this talk I will discuss the concept of irrationality and give a short overview of related results. I will then discuss the specialization of dominant maps in smooth families.

~~Wednesday 15 February 2023, 3pm~~. No seminar this week

**Wednesday 22 February 2023, 3pm. Speaker: Nicola Pagani (Liverpool)**** Moved to 15 March**

**Wednesday 1 March 2023, 3pm. Speaker: Liana Heuberger (Bath)**

Title: Q-Fano threefolds and how to construct them

Abstract: I will give a brief overview about the techniques involved in constructing Fano varieties using mirror symmetry, appearing in the works of Coates, Corti, Kasprzyk et al. I will then describe one of its more concrete incarnations, a method of "inverting" a toric degeneration called Laurent Inversion, which I have used to construct 100 deformation families of Q-Fano threefolds.

**Wednesday 8 March 2023, 3pm. Speaker: Farhad Babaee (Bristol) Postponed to Fall 2023**

~~Title: Some applications of Tropical Geometry in Complex Geometry~~

~~Abstract: In this talk, I will recall two important questions in Complex Analytic Geometry, namely, a strong version of the Hodge Conjecture for Positive Currents and the Equidistribution Conjecture of Dinh--Sibony. I will also explain how Tropical Geometry can provide insight into these questions. No background in these topics is assumed.~~

**Wednesday 15 March 2023, 3pm-5pm. Speaker: Renzo Cavalieri (Colorado State U.)**

Title: Pseudo-Stable Hodge Integrals

Abstract: The moduli space of curves (by which I mean its Deligne-Mumford compactification) is a well studied object in algebraic geometry. Mumford introduced the notion of tautological intersection theory to study a part of the intersection theory which is simple enough to be tractable, but rich enough to be meaningful. Hodge integrals are a class of tautological intersection numbers that arise from intersecting the Chern classes of the Hodge bundle and of the cotangent line bundles. In the first part of the talk I will introduce all these concepts and review some ”classical” structural results about Hodge integrals. When running the MMP on the moduli space of curves, after the first wall-crossing one sees the moduli space of pseudo-stable curves, which is the target of a birational regular morphism from the moduli space of curves. We investigate how the Hodge bundles on either side of this morphism are related, and how, correspondingly, there are very rich combinatorial relations between Hodge integrals and pseudo stable Hodge integrals. This talk is based on joint work with Gallegos, Ross, Wise, Van Over and on some of Matthew Williams’ doctoral work.