Search results for "Algebraic Geometry"

On the slope of hyperelliptic fibrations with positive relative irregularity

2016

Let $f:\, S \to B$ be a locally non-trivial relatively minimal fibration of hyperelliptic curves of genus $g\geq 2$ with relative irregularity $q_f$. We show a sharp lower bound on the slope $\lambda_f$ of $f$. As a consequence, we prove a conjecture of Barja and Stoppino on the lower bound of $\lambda_f$ as an increasing function of $q_f$ in this case, and we also prove a conjecture of Xiao on the ampleness of the direct image of the relative canonical sheaf if $\lambda_f<4$.

The Oort conjecture on Shimura curves in the Torelli locus of curves

2014

Oort has conjectured that there do not exist Shimura curves contained generically in the Torelli locus of genus-$g$ curves when $g$ is large enough. In this paper we prove the Oort conjecture for Shimura curves of Mumford type and Shimura curves parameterizing principally polarized $g$-dimensional abelian varieties isogenous to $g$-fold self-products of elliptic curves for $g&gt;11$. We also prove that there do not exist Shimura curves contained generically in the Torelli locus of hyperelliptic curves of genus $g&gt;7$. As a consequence, we obtain a finiteness result regarding smooth genus-$g$ curves with completely decomposable Jacobians, which is related to a question of Ekedahl and Serre.

Local Gromov-Witten invariants are log invariants

2019

We prove a simple equivalence between the virtual count of rational curves in the total space of an anti-nef line bundle and the virtual count of rational curves maximally tangent to a smooth section of the dual line bundle. We conjecture a generalization to direct sums of line bundles.

Modular Calabi-Yau threefolds of level eight

2005

In the studies on the modularity conjecture for rigid Calabi-Yau threefolds several examples with the unique level 8 cusp form were constructed. According to the Tate Conjecture correspondences inducing isomorphisms on the middle cohomologies should exist between these varieties. In the paper we construct several examples of such correspondences. In the constructions elliptic fibrations play a crucial role. In fact we show that all but three examples are in some sense built upon two modular curves from the Beauville list.

Arithmetic hyperbolicity and a stacky Chevalley-Weil theorem

2020

We prove an analogue for algebraic stacks of Hermite-Minkowski's finiteness theorem from algebraic number theory, and establish a Chevalley-Weil type theorem for integral points on stacks. As an application of our results, we prove analogues of the Shafarevich conjecture for some surfaces of general type.

Effectively Computing Integral Points on the Moduli of Smooth Quartic Curves

2016

We prove an effective version of the Shafarevich conjecture (as proven by Faltings) for smooth quartic curves. To do so, we establish an effective version of Scholl's finiteness result for smooth del Pezzo surfaces of degree at most four.

Projective Crystalline Representations of \'Etale Fundamental Groups and Twisted Periodic Higgs-de Rham Flow

2017

This paper contains three new results. {\bf 1}.We introduce new notions of projective crystalline representations and twisted periodic Higgs-de Rham flows. These new notions generalize crystalline representations of \'etale fundamental groups introduced in [7,10] and periodic Higgs-de Rham flows introduced in [19]. We establish an equivalence between the categories of projective crystalline representations and twisted periodic Higgs-de Rham flows via the category of twisted Fontaine-Faltings module which is also introduced in this paper. {\bf 2.}We study the base change of these objects over very ramified valuation rings and show that a stable periodic Higgs bundle gives rise to a geometric…

A generalization of Dembowski's theorem on semi-planes

1981

Divisible designs and groups

1992

We study (s, k, λ1, λ2)-translation divisible designs with λ1≠0 in the singular and semi-regular case. Precisely, we describe singular (s, k, λ1, λ2)-TDD's by quasi-partitions of suitable quotient groups or subgroups of their translation groups. For semi-regular (s, k, λ1, λ2)-TDD's (and, more general, for the case λ2>λ1) we prove that their translation groups are either Frobenius groups or p-groups of exponent p. Some examples are given for the singular, semi-regular and regular case.

Sur la r�gularit� de la fonction croissance d'une vari�t� riemannienne

1994

On etudie la differentiabilite de la fonction croissance d'une variete riemannienne complete. En general, elle a la meme regularite qu'une fonction concave: la derivee peut avoir des sauts pour lesquels on donne une formule. Dans le cas analytique reel, la fonction croissance est de classeC1. Un exemple montre qu'elle n'est pas necessairementC2. A titre d'application, nous construisons, pour toute variete ouverte paracompacteM et toute fonction croissantev de classeC1, une metrique continue de croissance egale av et une metrique de classeC∞ surM de croissance proche dev en topologieC1-fine.