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The linear subspace section of variety by specializations

In this article, we will prove the preservation of some properties of the generic linear subspace sections of nondegenerate varieties by specializations and ground forms of components of variety are conjugated. | JOURNAL OF SCIENCE OF HNUE Mathematical and Physical Sci. 2012 Vol. 57 No. 7 pp. 12-19 This paper is available online at http stdb.hnue.edu.vn THE LINEAR SUBSPACE SECTION OF VARIETY BY SPECIALIZATIONS Dam Van Nhi School for Gifted Students Hanoi National University of Education Abstract. In this paper we prove the preservation of some properties of the generic linear subspace sections of nondegenerate varieties by specializations. Keywords Specialization variety absolutely irreducible. 1. Introduction To study smooth curves one intersects the curve with a general hyperplane and studies the resulting finite set of points. The Harris s lemma 2 about a set of points in the uniform position has attracted much attention in algebraic geometry. This is a set of points of a projective space such that any two subsets each with the same cardinality have the same Hilbert function. We bring up the question of whether other uniformed position properties remain unchanged when a curve is intersected by any hyperplane. To answer this question we use the notation ground form which was given by E. Noether 8 B.L. van der Waerden 11 and specializations of modules and of graded modules which was given by D. V. Nhi and N. V. Trung 6 7 . In this article we will prove the preservation of some properties of the generic linear subspace sections of nondegenerate varieties by specializations and ground forms of components of variety are conjugated. Throughout this paper Ω will be the universal field which is algebraic and has an infinite degree of transcendence over an infinite ground field K. 2. Some results about specializations of modules Let K be an infinite ground field and K be the algebraic closure of K. Denote by u u1 u2 . . . us a system of s new indeterminates ui which are algebraically independent of K K u x and the polynomial ring K u1 . . . us x1 . . . xn . We shall Received September 28 2012. Accepted October 5 2012. 2000 Mathematics Subject Classification Primary 13A30 .