A Connectedness Theorem for Products of Weighted Projective Spaces

Abstract

We prove a connectedness result for products of weighted projective spaces.

Connectivity and a Problem of Formal Geometry

“…Let me explain why this theorem is an improvement of the connectivity result proved in [4]. By Theorem 3 below the connectivity result of [4] (see Theorem 1 below) is equivalent to saying that the ring K( f (X) / f (X)∩∆ P ) is a field and the subfield K( f (X)) is algebraically closed in K( f (X) / f (X)∩∆ P ), while the above theorem is equivalent to saying that the natural map K( f (X)) → K( f (X) / f (X)∩∆ P ) is an isomorphism.…”

“…Università degli Studi di Genova, Dipartimento di Matematica, Via Dodecaneso 35, 16146 Genova, Italy, e-mail: badescu@dima.unige.it On the other hand, in [1] and [5] the connectivity results of Fulton-Hansen [14] and Debarre [7] have been improved to get stronger conclusions involving the G3 condition of Hironaka-Matsumura [17] on the extension of formal-rational functions on X along f −1 (∆ P ) (see Definition 1 below). The aim of the present paper is to improve the connectivity result of [4] in the same spirit.…”

“…Precisely, our aim is to prove the following strengthening of the connectivity result of [4] (see Theorem 1 below), and, under a slightly stronger hypothesis, also a generalization of the main result of [5]:…”

“…The connectivity result of Fulton and Hansen has been generalized in various directions by Hansen [15], Faltings [11], [12], Debarre [7], [8], [9], Bȃdescu [1], [5], Bȃdescu-Repetto [4], and others. Lucian On the other hand, in [1] and [5] the connectivity results of Fulton-Hansen [14] and Debarre [7] have been improved to get stronger conclusions involving the G3 condition of Hironaka-Matsumura [17] on the extension of formal-rational functions on X along f −1 (∆ P ) (see Definition 1 below).…”

“…The connectivity result of Fulton and Hansen has been generalized in various directions by Hansen [15], Faltings [11], [12], Debarre [7], [8], [9], Bȃdescu [1], [5], Bȃdescu-Repetto [4], and others.…”