1,720,980 research outputs found
Fujita decomposition on families of abelian varieties
No full textLet F: V→ B be a smooth non-isotrivial 1 - dimensional family of complex polarized abelian varieties and Vb= F- 1(b) be the general fiber. Let F1⊂ R1F∗C be the associated Hodge bundle filtration, Fb1=H1.0(Vb). Under the assumption that the Fujita decomposition for F1 is non trivial, that is there is a non trivial flat sub-bundle 0 ≠ U⊂ F1, we show that Vb has non-trivial endomorphism: End(Vb) ≠ Z
Massey Products and Fujita decompositions on fibrations of curves
No full textLet f: S→ B be a fibration of curves and let f∗ωS/B= U⊕ A be the second Fujita decomposition of f. In this paper we study a kind of Massey products, which are defined as infinitesimal invariants by the cohomology of a curve, in relation to the monodromy of certain subbundles of U. The main result states that their vanishing on a general fibre of f implies that the monodromy group acts faithfully on a finite set of morphisms and is therefore finite. In the last part we apply our result in terms of the normal function induced by the Ceresa cycle. On the one hand, we prove that the monodromy group of the whole U of hyperelliptic fibrations is finite (giving another proof of a result due to Luo and Zuo). On the other hand, we show that the normal function is non torsion if the monodromy is infinite (this happens e.g. in the examples shown by Catanese and Dettweiler)
On Subfields of the Function Field of a General Surface in P3
No full textIn this paper, we study birational immersions from a very general smooth plane curve to a nonrational surface with pg =q =0 to treat dominant rational maps from a very general surface X of degree ≥ 5 in P3 to smooth projective surfaces Y. Based on the classification theory of algebraic surfaces, Hodge theory, and deformation theory, we prove that there is no dominant rational map from X to Y unless Y is rational or Y is birational to X
Projective structures and Hodge theory
No full textEvery compact Riemann surface X admits a natural projective structure pu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} as a consequence of the uniformization theorem. In this work we describe the construction of another natural projective structure on X, namely the Hodge projective structure ph\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document}, related to the second fundamental form of the period map. We then describe how projective structures correspond to (1, 1)-differential forms on the moduli space of projective curves and, from this correspondence, we deduce that pu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} and ph\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} are not the same structure
On rational maps from the product of two general curves
No full textThis paper treats dominant rational maps from the product of two very general curves to nonsingular projective surfaces. Combining the result in [5] we prove that the product of two very general curves of genus g ≥ 7 and g′ ≥ 3 does not admit dominant rational maps of degree > 1 if the image surface is non-ruled. We also treat the case of the 2-symmetric product of a curve
Vanishing cohomology on a double cover
No full textIn this paper, we prove the irreducibility of the monodromy action on the anti-invariant part of the vanishing cohomology on a double cover of a very general element in an ample hypersurface of a complex smooth projective variety branched at an ample divisor. As an application, we study dominant rational maps from a double cover of a very general surface (Formula presented.) of degree (Formula presented.) in (Formula presented.) branched at a very general quadric surface to smooth projective surfaces (Formula presented.). Our method combines the classification theory of algebraic surfaces, deformation theory, and Hodge theory
Hyperelliptic Jacobians and isogenies
Full text linkIn this note we mainly consider abelian varieties isogenous to hyperelliptic Jacobians. In the first part we prove that a very general hyperelliptic Jacobian of genus g≥4is not isogenous to a non-hyperelliptic Jacobian. As a consequence we obtain that the intermediate Jacobian of a very general cubic threefold is not isogenous to a Jacobian. Another corollary tells that the Jacobian of a very general d-gonal curve of genus g≥4is not isogenous to a different Jacobian.
In the second part we consider a closed subvariety Y⊂Agof the moduli space of principally polarized varieties of dimension g≥3. We show that if a very general element of Yis dominated by the Jacobian of a curve Cand dimY≥2g, then Cis not hyperelliptic. In particular, if the general element in Yis simple, its Kummer variety does not contain rational curves. Finally we show that a closed subvariety Y⊂Mgof dimension 2g−1such that the Jacobian of a very general element of Yis dominated by a hyperelliptic Jacobian is contained either in the hyperelliptic or in the trigonal locus
Polygonal cycles in higher Chow groups of Jacobians
No full textThe aim of this paper is to construct non-trivial cycles in the first higher Chow group of the Jacobian of a curve having special torsion points. The basic tool is to compute the analogue of the Griffiths' infinitesimal invariant of the natural normal function defined by the cycle as the curve moves in the corresponding moduli space. We prove also a Torelli-like theorem. The case of genus 2 is considered in the last section
Brill-Noether loci for divisors on irregular varieties
Full text linkWetakeupthestudyoftheBrill–NoetherlociWr(L,X):={η∈Pic0(X)|h0(L⊗η) ≥ r + 1}, where X is a smooth projective variety of dimension > 1, L ∈ Pic(X), and r ≥ 0 is an integer.
By studying the infinitesimal structure of these loci and the Petri map (defined in analogy with the case of curves), we obtain lower bounds for h0(KD), where D is a divisor that moves linearly on a smooth projective variety X of maximal Albanese dimension. In this way we sharpen the results of [Xi] and we generalize them to dimension > 2.
In the 2-dimensional case we prove an existence theorem: we define a Brill–Noether number ρ(C, r) for a curve C on a smooth surface X of maximal Albanese dimension and we prove, under some mild additional assumptions, that if ρ(C,r) ≥ 0 then Wr(C,X) is nonempty of dimension ≥ ρ(C,r).
Inequalities for the numerical invariants of curves that do not move linearly on a surface of maximal Albanese dimension are obtained as an application of the previous results
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