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arXiv · 1602.03588

Blowing Up Finitely Supported Complete Ideals in a Regular Local Ring

William Heinzer·Youngsu Kim·Matthew Toeniskoetter
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Abstract

Let I be a finitely supported complete m-primary ideal of a regular local ring (R, m). We consider singularities of the normalization of the blow-up Proj R[It] of I. A theorem of Lipman implies that the ideal I has a unique factorization as a star-product of special star-simple complete ideals with possibly negative exponents for some of the factors. If the normalization of the projective model Proj R[It] is regular, we prove that it is the regular model obtained by blowing up the finite set of base points of I. Extending work of Lipman and Huneke-Sally in dimension 2, we prove that every local ring S on the normalization of Proj R[It] that is a unique factorization domain is regular. Moreover, if dim S is at least 2 and S dominates R, then S is an infinitely near point to R, that is, S is obtained from R by a finite sequence of local quadratic transforms.

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William Heinzer, Youngsu Kim, Matthew Toeniskoetter. 2016-02-11. Blowing Up Finitely Supported Complete Ideals in a Regular Local Ring. https://arxiv.org/abs/1602.03588

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