Rational normal scrolls and the defining equations of Rees algebras

By Andrew R. Kustin, C. Polini, Bernd Ulrich. 2008

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Abstract 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 .

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/d66a261f493f755b5c42f94042de779826fc955e

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In Algebra and Number Theory

  1. Andrew R. Kustin
  2. C. Polini
  3. Bernd Ulrich

In Computational Theory and Mathematics

  1. Andrew R. Kustin
  2. C. Polini
  3. Bernd Ulrich

In Geometry and Topology

  1. Andrew R. Kustin
  2. C. Polini
  3. Bernd Ulrich

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