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We are grateful to the seven speakers who have kindly agreed to contribute to this meeting. Below, the titles and abstracts of their talks are listed in alphabetical order.
Thomas Blomme (Université de Neuchâtel)
Multiple cover formula, between decomposition and correlation
Thomas Blomme (Université de Neuchâtel)
Multiple cover formula, between decomposition and correlation
Title: Multiple cover formula, between decomposition and correlation
Abstract: Abelian surfaces are complex tori whose enumerative invariants seem to satisfy remarkable regularity properties. The computation of their reduced Gromov-Witten invariants has already achieved in the case of primitive classes (Bryan-Oberdieck-Pandharipande-Yin). G. Oberdieck conjectured a few years ago a multiple cover formula expressing in a very simple way the invariants for the non-primitive classes in terms of the primitive one. This would close the computation of GW invariants for abelian surfaces. In this talk, we aim to explain the conjecture and how to prove most of it without being able to compute a Gromov-Witten invariant. This is joint work in progress with Francesca Carocci.
Victor Chachay (Université de Bourgogne)
A motivic enrichment for the count of lines in a degree 2 del Pezzo surface
Victor Chachay (Université de Bourgogne)
A motivic enrichment for the count of lines in a degree 2 del Pezzo surface
Title: A motivic enrichment for the count of lines in a degree 2 del Pezzo surface
Abstract: The number of lines in a del Pezzo surface S is a classical invariant when we work over an algebraically closed field. To recover an invariant for every field, we translate the problem in the motivic intersection theory, thus replacing the Chow rings for Chow-Witt rings and the top Chern class by some Euler class. To define this generalized invariant of line, we will need to realize the Hilbert scheme of lines of S as the zero locus of a section of a vector bundle to compute its Euler class.
Pascal Fong (Leibniz Universität Hannover)
Automorphism groups of Mori del Pezzo fibrations
Pascal Fong (Leibniz Universität Hannover)
Automorphism groups of Mori del Pezzo fibrations
Title: Automorphism groups of Mori del Pezzo fibrations
Abstract: Building on the ideas of Blanc, Fanelli, and Terpereau [Connected algebraic groups acting on three-dimensional Mori fibrations, IMRN 2023, no. 2, 1572-1689] for recovering Umemura's classification of maximal connected algebraic subgroups of the space Cremona group, we study the automorphism groups of Mori del Pezzo fibrations over smooth projective curves of positive genus. This is a joint work with S. Zimmermann.
Daniela Paiva Peñuela (Universität Basel)
The generalized Gizatullin’s problem
Daniela Paiva Peñuela (Universität Basel)
The generalized Gizatullin’s problem
Title: The generalized Gizatullin’s problem.
Abstract: The problem of determining which automorphisms of a smooth quartic surface S ⊂ P3 are induced by birational maps of P3 remains open. This question is known as Gizatullin’s problem. In this talk, we will discuss this problem and a generalized version, where we pose the same question for projective K3 surfaces contained in Fano threefolds. I we will provide a general overview of the theory of K3 surfaces and the birational geometry of Fano threefolds, and explain how the interaction between these two areas can be used to approach Gizatullin’s problem. The results I will present are part of several joint works with Carolina Araujo, Michela Artebani, Alice Garbagnati, Ana Quedo, and Sokratis Zikas.
Keyao Peng (Université de Bourgogne)
Cellular A1-Homology of Smooth Toric Varieties
Keyao Peng (Université de Bourgogne)
Cellular A1-Homology of Smooth Toric Varieties
Title: Cellular A1-Homology of Smooth Toric Varieties
Abstract: In this talk, we explore the calculations of cellular A1-homology for smooth toric varieties, an analog of classic cellular homology. We provide an explicit description of pure shellable cases and discuss the derivation of the (Milnor-Witt) motivic decomposition for these cases, inspired by classic results in real toric manifolds. These findings offer new and refined algebraic invariants for toric varieties, reflecting both their complex and real points. Additionally, we can present an additive basis for the Chow groups of general smooth toric varieties.
Linus Erik Rösler (École Polytechnique Fédérale de Lausanne)
Seshadri constants of adjoint divisors on surfaces and threefolds in arbitrary characteristic
Linus Erik Rösler (École Polytechnique Fédérale de Lausanne)
Seshadri constants of adjoint divisors on surfaces and threefolds in arbitrary characteristic
Title: Seshadri constants of adjoint divisors on surfaces and threefolds in arbitrary characteristic
Abstract: It is a folklore conjecture that for a fixed dimension d, there exist positive integers e and f such that for all smooth projective varieties X of dimension d and ample divisors A on X, the linear system |eK_X+fA| is basepoint free. While this is known in characteristic zero and for d<=2, already the threefold case in positive characteristic is wide open. One of the most promising approaches is to find effective lower bounds of Seshadri constants of adjoint bundles. We develop a new approach in this direction, and as an application we recover the surface case and obtain effective rationality results of small Seshadri constants on threefolds.
Anne Schnattinger (Université de Neuchâtel)
Weak Fano threefolds arising as the blowup of a hyperquadric in P^4 along a curve
Anne Schnattinger (Université de Neuchâtel)
Weak Fano threefolds arising as the blowup of a hyperquadric in P^4 along a curve
Title: Weak Fano threefolds arising as the blowup of a hyperquadric in P^4 along a curve
Abstract: Given a smooth hyperquadric Y in P^4, we consider its blowup X along a smooth irreducible curve C contained in Y. We study the question, when X is a weak Fano threefold, that is, when it has a nef and big anticanonical divisor. We are able to give a complete classification of such threefolds only depending on some geometric properties of C, particularly its genus and degree. We introduce the main proof ideas which include the analysis of the linear system given by the anticanonical divisor of X, as well as studying curves contained in a smooth K3 surface of degree 6.
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