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Basel-Dijon-EPFL seminar
Basel, November 13-14, 2023
Schedule
In Kollegienhaus, Hörsaal 116 (Monday) and 119 (Tuesday). Petersgraben 50, 4051 Basel.
Monday
November 13
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Tuesday
November 14
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10h15-11h15
Matilde Maccan
Classifying rational projective homogeneous varieties in positive characteristic
11h30-12h30
Lucas Moulin
Real forms of almost homogeneous SL2-threefolds
14h15-15h15
Domenico Valloni
Rational points on the Noether-Lefschetz locus of K3 moduli spaces
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lunch |
13h30-14h30
Douglas Guimarães
Ulrich bundles on isotropic Grasmannians
15h00-16h00
Evgeny Shinder
Birational maps, motivic invariants and boundedness
16h30-17h30
Fabio Bernasconi
Rational points on 3-folds with nef anti-canonical class over finite fields
social dinner
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Fabio Bernasconi (Basel) - Rational points on 3-folds with nef anti-canonical class over finite fields
A theorem of Esnault states that smooth Fano varieties over finite fields have rational points. What happens if we relax the conditions related to the positivity properties of the anti-canonical class? In this seminar, I will discuss the case of 3-folds with nef anti-canonical class. Specifically, we show that in the case of negative Kodaira dimension, the existence of rational points is established if the cardinality is greater than 19. In the K-trivial case, we prove a similar result, provided that the Albanese morphism is non-trivial. This is joint work with S. Filipazzi.
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Douglas Guimarães (Dijon) - Ulrich bundles on isotropic Grasmannians
In this talk, we will give a brief introduction to Ulrich bundles. We will discuss the problem of showing the existence of Ulrich bundles in certain varieties. In particular, we will show examples of such bundles on partial flag varieties and isotropic Grasmannians, following the work of Costa et al, and Foranev. Finally, we will discuss a partial result of an ongoing project on isotropic Grasmannian.
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Matilde Maccan (Rennes) - Classifying rational projective homogeneous varieties in positive characteristic
Any rational homogeneous projective variety can be written as a quotient of a semisimple group by a so-called parabolic subgroup. In this talk we will complete the classification of parabolics and formulate it in a uniform way, independent of type and characteristic. The cases we focus on are of a base field of characteristic two or three. If time allows, we will mention the construction of an exotic G2-variety of Picard rank one, as well as a few geometric consequences.
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Lucas Moulin (Dijon) - Real forms of almost homogeneous SL2-threefolds
Given a complex algebraic variety X, a classical problem in algebraic geometry is to determine the real forms of X, where a real form is an algebraic variety Y such that Yℂ := Y ×Spec(ℝ) Spec(ℂ) ≃ X. We are interested in a slightly different case : The case of G-varieties, where G is a complex algebraic group, and their real forms which are equivariant under the action of a real algebraic group F such that Fℂ ≃ G.
After having defined and given some examples of those notions, we will consider the case of SL2-threefolds containing an open orbit, which we call almost homogeneous SL2-threefolds. These varieties are well known in algebraic geometry, they appear for example in the classification of Fano varieties or in the study of algebraic subgroups of the Cremona group. I will explain how we can classify their equivariant real forms and the link with non-equivariant real forms (meaning when we forget about the SL2-action).
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Domenico Valloni (EPFL) - Rational points on the Noether-Lefschetz locus of K3 moduli spaces
Let L be an even hyperbolic lattice and denote by $\mathcal{F}_L$ the moduli space of L-polarized K3 surfaces. In this talk, I will explain criterion to decide whether a given K-rational point of $\mathcal{F}_L$ has generic Néron-Severi lattice (that is, $\mathrm{NS}(X) ≃ L$), where K is any number field. As a consequence, I will show that the Bombieri-Lang conjecture implies non density statements for such rational points, as predicted by conjectures of Coleman and Shafarevich.
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Evgeny Shinder (Sheffield) - Birational maps, motivic invariants and boundedness
I will recall motivic invariants of birational maps, and introduce them in the relative setting, for varieties over a base. Among applications, I will explain (1) a new vanishing result and (2) unboundedness of centers, considered up to cancellation. This is joint work in progress with Hsueh-Yung Lin.
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Participants
Ahmed Abouelsaad (Basel)
Jérémy Blanc (Basel)
Fabio Bernasconi (Basel)
Anna Bot (Basel)
Victor Chachay (Dijon)
Adrien Dubouloz (Poitiers)
Mani Esna Ashari (Basel)
Daniele Faenzi (Dijon)
Douglas Guimarães (Dijon)
Archi Kaushik (EPFL)
Chrislaine Kuster (Dijon)
Matilde Maccan (Rennes)
Felipe Monteiro (Dijon)
Lucas Moulin (Dijon)
Lucy Moser-Jauslin (Dijon)
Alapan Mukhopadhyay (EPFL)
Aliaksandra Novik (EPFL)
Emre Ozavci (EPFL)
Linus Rosler (EPFL)
Anne Schnattinger (Basel)
Julia Schneider (U.Zürich)
Evgeny Shinder (Sheffield)
Nikolaos Tsakanikas (EPFL)
Christian Urech (Basel)
Immanuel van Santen (Basel)
Domenico Valloni (EPFL)
Henrik Wehrheim (Basel)
Aline Zanardini (EPFL)
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