The meeting will be held on the 28 February and 1 March in the Alan Turing Building

(directions) at the University of Manchester.
All talks take place in Frank Adams Room 1 on the first floor.

### Thursday, 28 February

### Friday, 1 March

Further to the talks, we will have a social dinner on Thursday evening.

#### Organisers:

Christopher Frei,

Daniel Loughran,

Hendrik Süß
#### Travel Expenses:

**We have funding available to cover travel and accommodation for participants, in particular PhD students and postdocs.**
Please contact Johan Martens (

johan.martens@ed.ac.uk).

The meeting is supported by the LMS.

### Abstracts

#### Yohan Brunebarbe: Algebraicity of period maps via o-minimal GAGA

We prove a conjecture of Griffiths on the quasi-projectivity of images
of period maps using algebraization results arising from o-minimal geometry (joint with Benjamin Bakker and Jacob Tsimerman).

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

A question of Orlov is whether the derived category of the Grothendieck--Knudsen moduli space of stable, rational curves with n markings admits a full, strong, exceptional collection that is invariant under the action of the symmetric group \(S_n\). I will present an approach towards answering this question. This is joint work with Jenia Tevelev.

#### Jesus Martinez Garcia: The moduli continuity method for log Fano pairs

The existence of a Kähler-Einstein metric on a

**Q**-Gorenstein
smoothable Fano variety is equivalent to the algebro-geometric property of
K-stability. It is natural to perturb the metric along a
pluri-anticanonical divisor, which gives rise to the equivalent property of
log K-stability for log pairs. The 'moduli continuity method' has permitted
to explicitly realise certain connected components of the moduli of
K-stable Fano varieties as GIT quotients. In this talk, we extend the
method to log pairs to study some fundamental examples, including log pairs
formed by a cubic surface and an anti-canonical divisor, and those formed
by projective space and hyperplane sections of high degree. As an
application, we show that the moduli of log pairs in these examples is
projective.

####
Simon Pepin Lehalleur: A formula for the motive of the moduli stack of vector bundles on a curve

Following Grothendieck's vision that a motive of an algebraic variety should capture many of its cohomological invariants, Voevodsky introduced a triangulated category of motives which partially realises this idea. After describing some properties of this category, I will explain how to define the motive of certain algebraic stacks. I will then state and sketch a proof for the motive of the moduli stack of vector bundles on a smooth projective curve; this formula is compatible with classical computations of invariants of this stack due to Harder, Atiyah-Bott and Behrend-Dhillon. The proof involves rigidifying this stack using Quot and Flag-Quot schemes parametrising Hecke modifications as well as a motivic version of an argument of Laumon and Heinloth on the relative cohomology of small maps. This is joint work with Vicky Hoskins.

#### Gregory Sankaran: Hyperelliptic genus 4 curves and abelian surfaces

An an ample line bundle L of type (d_1,d_2) on an abelian surface A provides a linear system |L| of curves of genus 1+d_1d_2. Surprisingly little is known about which curves actually arise this way. We approach this problem via the Jacobian of the curve and study in particular the case of polarisation (1,3), when there is a distinguished element of |L|. Using this, we show by geometric means that there is, up to translation, exactly one smooth hyperelliptic genus 4 curve on a general (1,3)-polarised abelian surface. This provides a geometrically descriptive explanation of a numerical computation by Bryan, Oberdieck, Pandharipande and Yin. This is joint work with Pawel Borowka (Krakow).

#### Susanna Zimmermann: Quotients of plane Cremona groups

The Cremona group is the group of binational maps of the projective space. I would like to explain how to construct a quotient of the plane Cremona group if the ground field has an extension of degree 8.