Measures of irrationality for projective varieties (Lecture 1)
Catégorie d'évènement : Groupe de travail Géométrie Date/heure : 20 novembre 2023 10:15-12:15 Lieu : Salle de conférences Nancy Oratrice ou orateur : Francesco Bastianelli Résumé :An important problem in algebraic geometry is understanding whether a given variety satisfies some rationality property, such as being rational, uniruled, rationally connected, unirational or stably rational.
The purpose of these lectures is to investigate a complementary circle of questions: in what manner can one quantify and control `how irrational’ a given complex projective variety X might be?
In this direction, we will consider various birational invariants called `measures of irrationality’, which somehow measure the failure of a given projective
variety to satisfy the rationality properties listed above. In particular, I will focus on the `degree of irrationality’ (i.e. the least degree of a dominant rational map from X to the projective space) and the `covering gonality’ (i.e. the least gonality of a curves passing through a general point of X).
Initially, I will introduce these ideas and I will discuss various examples and results. Then I will present some techniques based on positivity properties of canonical bundles, which lead to lower bounds for those birational invariants. Finally, I will show how these techniques can be used to describe the invariants for hypersurfaces of large degree and for other varieties of general type.
Main references:
– F. Bastianelli, R. Cortini and P. De Poi, The gonality theorem of Noether for hypersurfaces, J. Algebraic Geom. 23 (2014), 313-339.
– F. Bastianelli, P. De Poi, L. Ein, R. Lazarsfeld, B. Ullery, Measures of irrationality for hypersurfaces of large degree, Compos. Math. 153 (2017), 2368-2393.
– F. Bastianelli, Irrationality issues for projective surfaces, Boll. Unione Mat. Ital. 11 (2018), 13-25.
– F. Bastianelli, C. Ciliberto, F. Flamini, P. Supino, Gonality of curves on general hypersurfaces, J. Math. Pures Appl. 125 (2019), 94-118.
Cayley-Bacharach condition and applications
Catégorie d'évènement : Séminaire de géométrie complexe Date/heure : 20 novembre 2023 14:00-15:00 Lieu : Salle de conférences Nancy Oratrice ou orateur : Francesco Bastianelli Résumé :Let X be a complex projective variety and let |D| be a complete linear system on X.
In line with the famous Cayley-Bacharach theorem, we say that a finite set S of points of X satisfies the Cayley-Bacharach condition with respect to |D| if any effective divisor of |D| passing through all but one point of S, passes also through the last point.
In this talk, we report on a joint work with Nicola Picoco, and we discuss the Cayley-Bacharach condition on projective spaces and Grassmannians, along with some applications.
In particular, we describe how Cayley-Bacharach condition on the Grassmannian G(k,n) imposes restrictions on the linear span of the corresponding k-planes in the n-dimensional projective space.
Moreover, we apply our result to study the covering gonality of the k-fold symmetric product of a smooth curve C. Namely, we prove that if k=2,3,4, the covering gonality of the k-fold symmetric product equals the gonality of C, and we characterize families of curves computing this invariant when C is a general curve.
A lower bound of the first Steklov-Dirichlet eigenvalue for eccentric annuli
Catégorie d'évènement : Séminaire de géométrie différentielle Date/heure : 20 novembre 2023 15:30-16:30 Lieu : Oratrice ou orateur : Dong-Hwi Seo Résumé :The Steklov eigenvalue problem is an eigenvalue problem for an operator which is defined in the boundary of a domain. Since the operator is nonlocal, the eigenvalues depend on both the geometries of the interior and the boundary of the domain. In this talk, we consider the Steklov-Dirichlet eigenvalue problem in eccentric annuli and related problems. We obtain a lower bound of the first Steklov-Dirichlet eigenvalues of the eccentric annuli by analyzing the first eigenvalues if the distance between the boundary components are sufficiently close. This is based on joint work with Jiho Hong and Mikyoung Lim.