### Workshop in Bordeaux (March 2023)

#### Dates and places

Thursday 9 - Friday 10 March : Salle de Conférences, bâtiment A33, Institut de Mathématiques de Bordeaux.

#### Post-strike schedule

 Thursday 9 10:30-12:00 Welcome coffee 12:00-14:00 lunch break 14:30-15:30 Margaret Bilu Rational curves on hypersurfaces and a motivic circle method. 15:30-16:00 coffee break 16:00-17:00 Sokratis Zikas Unbounded connected algebraic subgroups of $Bir(C\times \mathbb{P}^n)$. Friday 10 9:00-9:15 Andrea Fanelli Short introduction to the FRACASSO Project 9:15-10:15 Cinzia Casagrande Fano 4-folds with $b_2>12$ are products of surfaces 10:15-10:45 coffee break 10:45-11:45 Nicolas Perrin Séminaire de Géométrie : VMRT of wonderful compactifications of symmetric spaces. 11:45-13:30 lunch break 13:30-14:30 Marta Pieropan Séminaire de Théorie des Nombres : On rationally connected varieties over $C_1$ fields of characteristic $0$

#### Titles and abstracts

• ##### Margaret Bilu (Bordeaux): Rational curves on hypersurfaces and a motivic circle method

The Hardy–Littlewood circle method is a well-known technique of analytic number theory that has successfully solved several major number theory problems. More recently, a version of the method over function fields, combined with spreading out techniques, has led to new results about the geometry of moduli spaces of rational curves on hypersurfaces of low degree. I will report on joint work with Tim Browning on implementing a circle method with an even more geometric flavour, where the computations take place in a suitable Grothendieck ring of varieties, leading thus to more precise description of the geometry of the above moduli spaces.

• ##### Cinzia Casagrande (Turin): Fano 4-folds with $b_2>12$ are products of surfaces

Let $X$ be a smooth, complex Fano 4-fold, and $b_2$ its second Betti number. We will discuss the following result: if $b_2>12$, then $X$ is a product of del Pezzo surfaces. The proof relies on a careful study of divisorial elementary contractions $f\colon X\to Y$ such that the image S of the exceptional divisor is a surface, together with my previous work on Fano 4-folds. In particular, given $f\colon X\to Y$ as above, under suitable assumptions we show that $S$ is a smooth del Pezzo surface with $-K_S$ given by the restriction of $-K_Y$.

• ##### Nicolas Perrin (École polytechnique): VMRT of wonderful compactifications of symmetric spaces.

(joint work with M. Brion and S. Kim) In this talk I will describe the VMRT (= Variety of Minimal Rational Tangents) of wonderful compactifications of symmetric spaces. In particular, we will prove that in (almost) all cases, there is a unique VMRT although these compactifications have a large Picard number. An important tool is the restricted root system which controls many important features of the geometry of wonderful compactifications.

• ##### Marta Pieropan (Utrecht): On rationally connected varieties over $C_1$ fields of characteristic 0

In the 1950s Lang studied the properties of $C_1$ fields, that is, fields over which every hypersurface of degree at most n in a projective space of dimension n has a rational point. Later he conjectured that every smooth proper rationally connected variety over a $C_1$ field has a rational point. The conjecture is proven for finite fields (Esnault) and function fields of curves over algebraically closed fields (Graber–Harris–de Jong–Starr), but it is still open for the maximal unramified extensions of $p$-adic fields. I use birational geometry in characteristic 0 to reduce the conjecture to the problem of finding rational points on Fano varieties with terminal singularities, and I provide some evidence in dimension 3.

• ##### Sokratis Zikas (Poitiers): Unbounded connected algebraic subgroups of $Bir(C\times \mathbb{P}^n)$.

The classification of maximal connected algebraic subgroups of the group of birational transformations $Bir(\mathbb{P}^m)$ of $\mathbb{P}^m$ for m = 2 and 3 implies that every connected algebraic subgroup of $Bir(\mathbb{P}^m)$ is contained in a maximal one. Thus a natural question is whether a similar statement is true for $Bir(C \times \mathbb{P}^n)$, where $C$ is a curve of positive genus. In this talk, I will give a negative answer to the previous question. The proof relies on the machinery of the $G$-equivariant Sarkisov Program. This is joint work with Pascal Fong.

#### Participants

• Margaret Bilu (Bordeaux)
• Cinzia Casagrande (Turin)
• Ana-Maria Castravet (Versailles)
• Loïs Faisant (Grenoble)
• Andrea Fanelli (Bordeaux)
• Enrica Floris (Poitiers)
• Jean-Philippe Furter (Bordeaux)
• Bianca Gouthier (Bordeaux)
• Qing Liu (Bordeaux)
• Sara Mehidi (Toulouse)
• Boaz Moerman (Utrecht)
• Duc-Manh Nguyen (Bordeaux)
• Nicolas Perrin (École polytechnique)
• Marta Pieropan (Utrecht)
• Ernest Speka (Versailles)
• Dajano Tossici (Bordeaux)
• Sokratis Zikas (Poitiers)

#### Organiser

Andrea Fanelli (Université de Bordeaux)

#### Funding

This meeting is supported by: