# Algebraic Geometry Seminar

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: TBA

Abstract: TBA

**Wednesday 15 February 2023, 3pm. Speaker: TBA**

Title: TBA

Abstract: TBA

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

Title: TBA

Abstract: TBA

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

Title: TBA

Abstract: TBA

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

Title: TBA

Abstract: TBA

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

Title: TBA

Abstract: TBA