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A non-divisorial variety Fraga, Robert Joseph

Abstract

When divisorial varieties were first introduced, the question immediately arose whether there are any varieties which are not divisorial. This work answers the question in the affirmative. We prove here that the non-projective variety M defined by Nagata in Memoirs of the College of Science, University of Kyoto, Series A, Vol. XXX, Mathematics No. 3, 1957, pp. 231-235 is, in fact, non-divisorial. The work is organized as follows: We first discuss briefly the concepts relating to the notion of divisorial variety. Next there is a description of Nagata's variety in which we include the proofs of statements which we shall need for the subsequent theorems. The preliminary results are of two types: first we prove several lemmas concerning the dominance of local rings of points on the variety M. Second we prove that divisors whose varieties contain the vertex of the affine cone V used in Nagata's example must intersect the line at infinity of the cone at a point(s) whose local ring dominates the local ring of the vertex of the cone Vσ under the transformation σ defined by Nsgata. This result indicates strongly that the variety M = VU Vσ is not divisorial. For the proof that this is, in fact, the case, we prove in detail a strictly algebraic result to the effect that the (prime) ideals associated with the irreducible components of a divisor which do not contain the vertex P of the cone V are principal. With this result, we finally show by contradiction that M is not divisorial.

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