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Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Cuntz, Joachim / Huybrechts, Daniel / Hwang, Jun-Muk

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Rational normal scrolls and the defining equations of Rees algebras

1Mathematics Department, University of South Carolina, Columbia, SC 29208

2Mathematics Department, University of Notre Dame, Notre Dame, IN 46556

3Department of Mathematics, Purdue University, West Lafayette, IN 47907

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal). Volume 2011, Issue 650, Pages 23–65, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: 10.1515/crelle.2011.002, January 2011

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Consider a height two ideal, I, which is minimally generated by m homogeneous forms of degree d in the polynomial ring R = k[x, y]. Suppose that one column in the homogeneous presenting matrix φ of I has entries of degree n and all of the other entries of φ are linear. We identify an explicit generating set for the ideal which defines the Rees algebra ℛ = R[It]; so for the polynomial ring S = R[T 1, . . . , Tm]. We resolve ℛ as an S-module and Is as an R-module, for all powers s. The proof uses the homogeneous coordinate ring, A = S/H, of a rational normal scroll, with . The ideal is isomorphic to the n th symbolic power of a height one prime ideal K of A. The ideal K (n) is generated by monomials. Whenever possible, we study A/K (n) in place of because the generators of K (n) are much less complicated then the generators of . We obtain a filtration of K (n) in which the factors are polynomial rings, hypersurface rings, or modules resolved by generalized Eagon–Northcott complexes. The generators of I parameterize an algebraic curve in projective m – 1 space. The defining equations of the special fiber ring ℛ/(x, y)ℛ yield a solution of the implicitization problem for .

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