Fano varieties: RAtional Curves, Arithmetic, Sections over Surfaces and Obstructions

• Marcello Bernardara (Toulouse): Conic Bundles of K3 type and Hyperkähler manifolds.

  Cubic and Gushel-Mukai fourfolds carry (1 and 2 respectively) conic bundle structures, whose discriminants are nodal surfaces whose double covers are of general type. The anti-invariant part of the intermediate cohomology of the latter surfaces carries the K3 structure corresponding to the one in the intermediate cohomology of the fourfolds. In a work in collaboration with Fatighenti, G. Kapustka, M. Kapustka, Manivel, Mongardi and Tanturri, we prove that in the case of Gushel-Mukai fourfolds, the discriminant double covers can be described as sections of HK manifolds with an anti-symplectic involution. Moreover, we analyze 3 other families of Fano fourfolds of K3 type with conic bundles degenerating along the same discriminants as above and we relate each family to one of the previous via hyperbolic splitting.

• Alexandru Dimca (Nice): Rational curves and free curves in the plane

  This will be a survey talk about the relations between the rational cuspidal plane curves and the free curves. The final part, which is new, will discuss a topic which was of great interest in Bordeaux some 30 years ago.

• Tiago Duarte Guerreiro (Orsay): On hypersurfaces in projective bundles

  Mori dream spaces are a special kind of varieties introduced by Hu and Keel in 2000 that enjoy very good properties with respect to the minimal model program. On the other hand, not many classes of examples of these are known. In this talk we introduce general hypersurfaces in certain projective bundles of Picard rank 2 and show that (some of) these are Mori dream spaces, partially generalising Ottem's result on hypersurfaces in products of projective spaces.

• Jean-Philippe Furter (IMB): Families of Cremona transformations

  A family of Cremona transformations is a « morphism » from an algebraic variety to the Cremona group. Such « morphisms" have been defined by Demazure and Serre. In this talk, we will give a few examples of families of Cremona transformations and we will also give some general properties of them (for example related with the length, the dynamical degree, the homaloidal type).

• Jesus Martinez-Garcia (Essex): Variations of K-moduli for del Pezzo surfaces

  Ascher, DeVleming and Liu constructed a theory of variations of K-moduli of log Fano pairs, in which the coefficients of the divisors are allowed to change, introducing birational transformations on the K-moduli. The most natural example is K-moduli of smoothable log del Pezzo pairs formed by a del Pezzo surface and an anti-canonical divisor, a natural generalisation of the first description of K-moduli for del Pezzo surfaces given by Odaka-Spotti-Sun. Our case also implies analytic questions previously considered by Szekelyhidi on the existence of Kahler-Einstein metrics with conical singularities along a divisor on del Pezzo surfaces. For degrees 2, 3 and 4 we establish an isomorphism between the K-moduli spaces and variation of Geometric Invariant Theory compactifications. For degrees 2-9, we describe the wall-chamber structure of the K-moduli of these problems, including all K-polystable replacements. This is joint work with Theodoros Papazachariou and Junyan Zhao.

• Mirko Mauri (École polytechnique): An algebro-geometric version of the Poincaré conjecture

  Dual complexes are CW-complexes, encoding the combinatorial data of how the irreducible components of a simple normal crossing pair intersect. The algebraic geometry of the divisor is reflected in the topology of the dual complex. One of the most tantalizing conjectures in the field is the expectation that the dual complex of an anticanonical divisor is a sphere or a finite quotient of a sphere. Equivalently, a combinatorial Calabi-Yau variety should resemble a sphere. I will provide an overview of the current state of the conjecture, and report on joint work with Joaquin Moraga. We introduce a numerical invariant called birational complexity that, among other properties, measures the degree to which the dual complex of an anticanonical divisor is close to be a sphere.

• Enrica Mazzon (Regensburg): Higher Fano manifolds

  Fano manifolds are complex projective manifolds having positive first Chern class. The positivity condition on the first Chern class has far-reaching geometric and arithmetic implications. For instance, Fano manifolds are covered by rational curves, and families of Fano manifolds over one-dimensional bases always admit holomorphic sections. In recent years, there has been a great effort toward defining suitable higher analogues of the Fano condition, which are expected to enjoy stronger versions of several of the nice properties of Fano manifolds. In this talk, I will discuss higher Fano manifolds which are defined in terms of positivity of higher Chern characters. After a brief survey of what is currently known, I will focus on higher Fano manifolds which are toric varieties. This is joint work with Carolina Araujo, Roya Beheshti, Ana-Maria Castravet, Kelly Jabbusch, Svetlana Makarova, and Nivedita Viswanathan.

• Dajano Tossici (IMB): Infinitesimal unipotent subgroup schemes of $PGL_2$ in characteristic 2

  In this talk we give a classification, up to isomorphism, of infinitesimal unipotent subgroup schemes of PGL_2 in characteristic 2. The situation is extremely different from characteristic different from 2, where any such a group scheme is isomorphic to a subgroup scheme of the additive group. This is a joint work with Bianca Gouthier.

Organiser

Andrea Fanelli (Université de Bordeaux)

Funding

This meeting is supported by:

• ANR FRACASSO