1,720,963 research outputs found
Corrigendum and Addendum to “Polarized Parallel Transport and Uniruled Divisors on Generalized Kummer Varieties”
Full text linkWe 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)
Density of Noether-Lefschetz loci of polarized irreducible holomorphic symplectic varieties and applications
Full text linkIn this paper, we derive from deep results due to Clozel and Ullmo a sharp density result of Noether-Lefschetz loci inside the moduli space of marked (polarized) irreducible holomorphic symplectic (IHS) varieties. In particular, we obtain the density of Hilbert schemes of points on projective K3 surfaces and of projective generalized Kummer varieties in their moduli spaces. We present applications to the existence of rational curves on projective deformations of such varieties, to the study of the Mori cone of curves and of the associated extremal birational contractions, and a refinement of Hassett's result on cubic fourfolds whose Fano variety of lines is isomorphic to a Hilbert scheme of two points on a K3 surface. We also discuss Voisin's conjecture on the existence of coisotropic subvarieties on IHS varieties and relate it to a stronger statement on Noether-Lefschetz loci in their moduli spaces
Regenerations and applications
Full text linkChen-Gounelas-Liedtke recently introduced a powerful regeneration technique, a process opposite to specialization, to prove existence results for rational curves on projective surfaces. We show that, for projective irreducible holomorphic symplectic manifolds, an analogous regeneration principle holds and provides a very flexible tool to prove existence of uniruled divisors, significantly improving known results
Singular curves on a K3 surface and linear series on their normalizations.
No full textWe 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
Families of rational curves on holomorphic symplectic varieties and applications to 0-cycles
Full text linkWe study families of rational curves on irreducible holomorphic symplectic varieties. We give a necessary and sufficient condition for a sufficiently ample linear system on a holomorphic symplectic variety of K3[n]-type to contain a uniruled divisor covered by rational curves of primitive class. In particular, for any fixed n, we show that there are only finitely many polarization types of holomorphic symplectic variety of K3[n]-type that do not contain such a uniruled divisor. As an application we provide a generalization of a result due to Beauville-Voisin on the Chow group of 0-cycles on such varieties
The Morrison–Kawamata cone conjecture for singular symplectic varieties
No full textWe prove the Morrison-Kawamata cone conjecture for projective primitive symplectic varieties with Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document}-factorial and terminal singularities with b2 >= 5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document}, from which we derive for instance the finiteness of minimal models of such varieties, up to isomorphisms. To prove the conjecture we establish along the way some results on the monodromy group which may be interesting in their own right, such as the fact that reflections in prime exceptional divisors are integral Hodge monodromy operators which, together with monodromy operators provided by birational transformations, yield a semidirect product decomposition of the monodromy group of Hodge isometries
Deformations of rational curves on primitive symplectic varieties and applications
Full text linkWe 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
Nodal Curves with General Moduli on K3 Surfaces
No full textWe 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).
No full textFor 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]
Nodal curves with general moduli on K3 surfaces
Full text linkWe 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 -nodal curve X sitting on a primitively polarized K3 surface S of degree 2p-2, for 2g=p-p11. 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
- …
