TAILIEUCHUNG - Báo cáo " THE HYPERSURFACE SECTIONS AND POINTS IN UNIFORM POSITION "

The aim of this paper is to show that the preservation of irreducibility of sections between a variety and hypersurface by specializations and almost all sections between a linear subspace of dimension h = n − d of Pn and a nondegenerate variety k of dimension d 0 consists of s points in uniform position. Introduction The lemma of Haaris [2] about a set in the uniform position has attracted much attention in algebraic geometry. That is a set of points of a projective space such that any two subsets of them with the same cardinality have the same. | VNU. JOURNAL OF SCIENCE Mathematics - Physics. N04 - 2005 THE HYPERSURFACE SECTIONS AND POINTS IN UNIFORM POSITION Pham Thi Hong Loan Pedagogical College Lao Cai Vietnam Dam Van Nhi Pedagogical University Ha Noi Vietnam Abstract. The aim of this paper is to show that the preservation of irreducibility of sections between a variety and hypersurface by specializations and almost all sections between a linear subspace of dimension h n d of P and a nondegenerate variety of dimension d 0 consists of s points in uniform position. Introduction The lemma of Haaris 2 about a set in the uniform position has attracted much attention in algebraic geometry. That is a set of points of a projective space such that any two subsets of them with the same cardinality have the same Hilbert function. For wider applicability of the result in this paper we will now apply this lemma to prove that almost all n d-dimensional linear subspace sections of a d-dimensional irreducible nondegenerate variety in Pn are the finite sets of points in uniform position under certain conditions. Here we use a notion ground-form which was given by E. Noether see 3 or 6 and specializations of ideals and of modules 3 4 5 6 7 that is a technique to prove the existence of algebraic structures over a field with prescribed properties. Let k be an infinite field of arbitrary characteristic. Let u 1 . um be a family of indeterminates and a a1 . am a family of elements of k. We denote the polynomial rings in n variables x-1 . xn over k u and k a by R k u x and by Ra k a x respectively. The theory of specialization of ideals was introduced by W. Krull 3 . Let I be an ideal of R. A specialization of I with respect to the substitution u a was defined as the ideal Ia f a x f u x E I n k u x . For almost all the substitutions u a that is for all a lying outside a proper algebraic subvariety of km specializations preserve basic properties and operations on ideals and the ideal Ia inherits most of the basic .

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