1,720,986 research outputs found
Quartic surfaces, their bitangents and rational points
No full textLet X be a smooth quartic surface not containing lines, defined over a number field κ. We prove that there are only finitely many bitangents to X which are defined over κ. This result can be interpreted as saying that a certain surface, having vanishing irregularity, contains only finitely many rational points. In our proof, we use the geometry of lines of the quartic double solid associated to X. In a somewhat opposite direction, we show that on any quartic surface X over a number field κ, the set of algebraic points in X(κ) which are quadratic over a suitable finite extension κ' of κ is Zariski-dense
The surface of Gauss double points
No full textWe study the surface of Gauss double points associated to a very general quartic surface and the natural morphisms associated to it
On integral points of some Fano threefolds and their Hilbert schemes of lines and conics
Full text linkWe prove some density results for integral points on affine open sets of Fano threefolds. For instance, let Xo= P3\ D where D is the union of two quadrics such that their intersection contains a smooth conic, or the union of a smooth quadric surface and two planes, or the union of a smooth cubic surface V and a plane Π such that the intersection V∩ Π contains a line. In all these cases we show that the set of integral points of Xo is potentially dense. We apply the above results to prove that integral points are potentially dense in some log-Fano or in some log-Calabi-Yau threefold
INTERSECTIONS IN SUBVARIETIES OF Glm AND APPLICATIONS TO LACUNARY POLYNOMIALS
No full textWe investigate intersections of a given subvariety X of Glm with cosets of 1-parameter subtori, on interpreting the context in terms of S-unit points over function fields. On adopting a function field version of a method introduced recently by the second author, extending to arbitrary dimensions previous work of the first and third authors, we prove that when the number of intersections is substantially higher than expected, one can classify the relevant subtori. As a consequence, we obtain a classification of the cosets of subtori such that there are many multiple intersections with X. This also allows a new proof of a conjecture of Erdős and Rényi on lacunary polynomials. We finally show how the methods yield results in the realm of Unlikely Intersections in Glm, and in the last section, reinterpret some of the results in terms of Vojta’s conjecture with truncated counting functions
BITANGENTS TO A QUARTIC SURFACE AND INFINITESIMAL TORELLI
No full textWe prove that the Hilbert scheme which parametrises bitangent lines to a general quartic surface is a smooth regular surface with no rational curves and with very ample canonical divisor. We also prove that it is a counterexample to infinitesimal Torelli and that its infinitesimal deformation space has dimension 20
Analytic and rational sections of relative semi-abelian varieties
Full text linkThe hyperbolicity statements for subvarieties and complement of hypersurfaces in abelian varieties admit arithmetic analogues, due to Faltings, Ann. Math. 133 (1991) (and for the semi-abelian case, Vojta, Invent. Math. 126 (1996); Amer. J. Math. 121 (1999)). In Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 29 (2018) by the second author, an analogy between the analytic and arithmetic theories was shown to hold also at proof level, namely in a proof of Raynaud’s theorem (Manin–Mumford Conjecture). The first aim of this paper is to extend to the relative setting the above mentioned hyperbolicity results. We shall be concerned with analytic sections of a relative (semi-)abelian scheme A → B over an affine algebraic curve B. These sections form a group; while the group of the rational sections (the Mordell–Weil group) has been widely studied, little investigation has been pursued so far on the group of the analytic sections. We take the opportunity of developing some basic structure of this apparently new theory, defining a notion of height or order functions for the analytic sections, by means of Nevanlinna theory
On certain permutation groups and sums of two squares
Full text linkWe consider the question of existence of ramified covers over P_1 matching certain prescribed ramification conditions. This problem has already been faced in a number of papers, but we discuss alternative approaches for an existence proof, involving elliptic curves and universal ramified covers with signature. We also relate the geometric problem with finite permutation groups and with the Fermat-Euler Theorem on the representation of a prime as a sum of two squares
The Betti map associated to a section of an abelian scheme
Full text linkGiven a point ξ on a complex abelian variety A, its abelian logarithm can be expressed as a linear combination of the periods of A with real coefficients, the Betti coordinates of ξ. When (A, ξ) varies in an algebraic family, these coordinates define a system of multivalued real-analytic functions. Computing its rank (in the sense of differential geometry) becomes important when one is interested about how often ξ takes a torsion value (for instance, Manin’s theorem of the kernel implies that this coordinate system is constant in a family without fixed part only when ξ is a torsion section). We compute this rank in terms of the rank of a certain contracted form of the Kodaira–Spencer map associated to (A, ξ) (assuming A without fixed part, and Zξ Zariski-dense in A), and deduce some explicit lower bounds in special situations. For instance, we determine this rank in relative dimension ≤ 3 , and study in detail the case of jacobians of families of hyperelliptic curves. Our main application, obtained in collaboration with Z. Gao, states that if A→ S is a principally polarized abelian scheme of relative dimension g which has no non-trivial endomorphism (on any finite covering), and if the image of S in the moduli space Ag has dimension at least g, then the Betti map of any non-torsion section ξ is generically a submersion, so that ξ-1Ators is dense in S(C)
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