Abstract We study integral points on varieties with infinite étale fundamental groups. More precisely, for a number field $F$ and $X/F$ a smooth projective variety, we prove that for any geometrically Galois cover $\varphi \colon Y \to X$ of degree at least $2\dim (X)^{2}$, there exists an ample line bundle $\mathscr{L}$ on $Y$ such that for a general member $D$ of the complete linear system $|\mathscr{L}|$, $D$ is geometrically irreducible and any set of $\varphi (D)$-integral points on $X$ is finite. We apply this result to varieties with infinite étale fundamental group to give new examples of irreducible, ample divisors on varieties for which finiteness of integral points is provable.
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Positivity of divisors on blown-up projective spaces, I
We study l-very ample, ample and semi-ample divisors on the blown-up projective space P^n in a collection of points in general position. We establish Fujita’s conjectures for all ample divisors with the number of points bounded above by 2n and for an infinite family of ample divisors with an arbitrary number of points.
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- Award ID(s):
- 2152130
- PAR ID:
- 10514986
- Publisher / Repository:
- Scuola Normale Superiore - Edizioni della Normale
- Date Published:
- Journal Name:
- Annali della Scuola normale superiore di Pisa Classe di scienze
- ISSN:
- 0391-173X
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Abstract Projective varieties with ample cotangent bundle satisfy many notions of hyperbolicity, and one goal of this paper is to discuss generalizations to quasi-projective varieties. A major hurdle is that the naive generalization is false—the log cotangent bundle is never ample. Instead, we define a notion called almost ample that roughly asks that it is as positive as possible. We show that all subvarieties of a quasi-projective variety with almost ample log cotangent bundle are of log general type. In addition, if one assumes globally generated then we obtain that such varieties contain finitely many integral points. In another direction, we show that the Lang–Vojta conjecture implies the number of stably integral points on curves of log general type, and surfaces of log general type with almost ample log cotangent sheaf are uniformly bounded.
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Abstract In this paper, we study {\mathbb{A}^{1}} -connected varieties from log geometry point of view, and prove a criterion for {\mathbb{A}^{1}} -connectedness. As applications, we provide many interesting examples of {\mathbb{A}^{1}} -connected varieties in the case of complements of ample divisors, and the case of homogeneous spaces. We also obtain a logarithmic version of Hartshorne conjecture characterizing projective spaces and affine spaces.more » « less
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null (Ed.)Abstract We continue our study, initiated in [34], of Riemann surfaces with constant curvature and isolated conic singularities. Using the machinery developed in that earlier paper of extended configuration families of simple divisors, we study the existence and deformation theory for spherical conic metrics with some or all of the cone angles greater than $2\pi $. Deformations are obstructed precisely when the number $2$ lies in the spectrum of the Friedrichs extension of the Laplacian. Our main result is that, in this case, it is possible to find a smooth local moduli space of solutions by allowing the cone points to split. This analytic fact reflects geometric constructions in [37, 38].more » « less
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This article is motivated by the authors’ interest in the geometry of the Mori dream space P^4 blown up in 8 general points. In this article, we develop the necessary technique for determining Weyl orbits of linear cycles for the four-dimensional case, by explicit computations in the Chow ring of the resolution of the standard Cremona transformation. In particular, we close this paper with applications to the question of the dimension of the space of global sections of effective divisors having at most 8 points.more » « less
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