Uniruledness of stable base loci of adjoint linear systems with and without Mori Theory

We explain how to deduce from recent results in the Minimal Model Program a general uniruledness theorem for base loci of adjoint divisors. We also show how to recover special cases by extending a technique introduced by Takayama.Comment: version 2 : improved exposition ; relaxed hypotheses on singularitie

Deformations of rational curves on primitive symplectic varieties and applications

We study the deformation theory of rational curves on primitive symplectic varieties and show that if the rational curves cover a divisor, then, as in the smooth case, they deform along their Hodge locus in the universal locally trivial deformation. As applica-tions, we extend Markman's deformation invariance of prime exceptional divisors along their Hodge locus to this singular framework and provide existence results for uniruled ample divisors on primitive symplectic varieties that are locally trivial deformations of any moduli space of semistable objects on a projective K3 or fibers of the Albanese map of those on an abelian surface. We also present an application to the existence of prime exceptional divisors

Hyperbolicity of varieties of log general type

These notes provide an overview of various notions of hyperbolicity for varieties of log general type from the viewpoint of both arithmetic and birational geometry. The main results are based on our paper entitled "Hyperbolicity and uniformity of varieties of log general type." They are expanded notes from a minicourse the authors gave as part of the Geometry and arithmetic of orbifolds workshop at UQ\'AM.Comment: Addressed some inaccuracies and typos pointed out by the referees and some readers. Slight change of title. To appear in CRM short courses (Arithmetic Geometry of Logarithmic Pairs and Hyperbolicity of Moduli Spaces: Hyperbolicity in Montr\'eal

Erratum: Polarized Parallel Transport and Uniruled Divisors on Generalized Kummer Varieties (International Mathematics Research Notices DOI: 10.13039/100009112)

We correct the statement of the main result of [9] and provide some further precisions. The goal of this short note is to state correctly the main result of [9]. For the definitions, the notations and the motivations we refer the reader to [9]. The correct statement is the following: Theorem 0.1. Let n = 1 be an integer (Formula Presented)

Higher dimensional tautological inequalities and applications

Let (X,F) be a smooth complex projective variety of dimension n endowed with a codimension 1 (possibly) singular foliation. Inspired by the celebrated Green-Griffiths conjecture and the results of McQuillan in the case of surfaces, the authors consider the problem of finding a suitable positivity condition on the foliated canonical bundle KF ensuring algebraic degeneracy of holomorphic maps f:Cn−1→X that are tangent to F. They obtain several results in that direction under various assumptions on the singularities of the foliation and on the positivity of KF and KX. For instance, when X has dimension 3, they are able to show that if KF is big and the singularities of F are canonical then there exists a proper algebraic subvariety Y⊂X containing all images of holomorphic maps f:C2→X tangent to the foliation. Their strategy is based on McQuillan's approach. They associate to any transcendental map f as above a positive closed (1,1)-current Tf. The core of their analysis is then to prove adequate tautological inequalities that give information on the positivity of the numerical class of the current Tf and of its lift to a suitable Grassmann bundle

Singular curves on a K3 surface and linear series on their normalizations.

We study the Brill-Noether theory of the normalizations of singular,irreducible curves on a K3 surface. We introduce a singular Brill-Noether number rho_sing and show that if Pic(K3) = Z[L], there are no linear series of degree d and dimension r on the normalizations of irreducible curves in |L|, provided that rho_sing < 0. We give examples showing the sharpness of this result. We then focus on the case of hyperelliptic normalizations, and classify linear systems |L| containing irreducible nodal curves with hyperelliptic normalizations, for rho_sing < 0, without any assumption on the Picard group

Nodal Curves with General Moduli on K3 Surfaces

We investigate the modular properties of nodal curves on a low genus K3 surface. We prove that a general genus g curve C is the normalization of a δ-nodal curve X sitting on a primitively polarized K3 surface S of degree 2p - 2, for 2 ≤ g = p - δ < p ≤ 11. The proof is based on a local deformation-theoretic analysis of the map from the stack of pairs (S, X) to the moduli stack of curves Mg that associates to X the isomorphism class [C] of its normalization

On families of rational curves in the Hilbert square of a surface (with an appendix by Edoardo Sernesi).

For any smooth surface S, the Hilbert scheme S^[n] parameterizing 0-dimensional length-n subschemes of S is a smooth 2n-dimensional variety whose inner geometry is naturally related to that of S. For instance, if E ⊂ S^[n] is the exceptional divisor—that is, the exceptional locus of the Hilbert–Chow morphism μ: S^[n] -> Sym^n(S) — then irreducible (possibly singular) rational curves not contained in E roughly correspond to irreducible (possibly singular) curves on S with a linear series of degree k and dimension 1 on their normalizations, for some k ≤ n. One of the features of this paper is to show how ideas and techniques from one of the two sides of the correspondence make it possible to shed light on problems naturally arising on the other side. If S is moreover a K3 surface then S^[n] is a hyperkähler manifold, and rational curves play a fundamental role in the study of the (birational) geometry of S^[n]

Análisis de polimorfismo de nucleótido simple en el gen FASN y su relación con características de producción en pollos

El objetivo de este trabajo es evaluar la posible asociación entre un polimorfismo de nucleótido simple (SNP) del gen FASN y cada uno de los siguientes caracteres productivos: Consumo de Alimento (CA), Ganancia de Peso (GP) y Peso Vivo Final (PVF). Durante la experiencia en el Instituto Nacional de Tecnología Agropecuaria (INTA), se organizaron y criaron 229 pollos de familias de hermanos enteros y de medios hermanos paternos. Cada una de las aves fue alojada en jaulas individuales con agua y alimentada con pellet ad libitum. El peso corporal se registró a los 54 días de edad y luego se determinó semanalmente y en forma individual el consumo de alimento y el peso durante 21 días. Los genotipos del gen FASN fueron identificados por amplificación por PCR y digeridos por la endonucleasa Hae III. La información fenotípica fue analizada en forma independiente por ANOVA con un modelo aleatorio. No se ha demostrado que los SNP evaluados en este trabajo afecten los caracteres analizados