Seshadri constants, diophantine approximation, and Roth’s theorem for arbitrary varieties

Abstract

Abstract. In this paper, we associate an invariant α x (L) to an algebraic point x on an algebraic variety X with an ample line bundle L. The invariant α measures how well x can be approximated by rational points on X, with respect to the height function associated to L. We show that this invariant is closely related to the Seshadri constant ǫ x (L) measuring local positivity of L at x, and in particular that Roth's theorem on P 1 generalizes as an inequality between these two invariants valid for arbitrary projective varieties.

Diophantine Approximation Constants for Varieties over Function Fields

“…We then use this result, Corollary 5.4, to prove Roth type theorems in a manner similar to what is done in [10]. and place what we do here in its proper context let us describe the results of [10] in some detail. To this end, let K be a number field, K an algebraic closure of K, X an irreducible projective variety defined over K, and x ∈ X(K).…”

“…The aim of this article is to study the complexity of approximating rational points of a projective variety defined over a function field of characteristic zero. Our motivation is work of McKinnon-Roth [10] and our main results, which we state in §1.2, show how the subspace theorem can be used to prove Roth type theorems, by analogy with those formulated in the number field setting, see [10, p. 515].…”

An Effective Suppression Method for 802.11 Transmission with Experimental Verification

“…We fix a closed point ξ ∈ X, and let π : X → X be the blow-up at ξ, and E = π −1 (ξ) be the exceptional divisor. We define the Seshadri constant of L at ξ as (26) ǫ(X, L; ξ) = ǫ(L, ξ) = sup{ǫ > 0| π * L − ǫE is nef }.…”

“…Remark 4.9. -By[26, Corollary 4.2], when X ֒→ P(E K ) is of degree δ with respect to O(1), we have the following lower bound of I X (H, ξ) introduced in Definition 4.6, which is…”

Distribution locale des points rationnels de hauteur bornée sur une surface de del Pezzo de degré 6

“…La démonstration est essentiellement celle de ( [5], Proposition 2.11) en remarquant que L est ample et donc vérifie la propriété de Northcott.…”