Rational normal scrolls and the defining equations of Rees algebras
-
Andrew R. Kustin
,
Claudia Polini
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[T1, . . . , 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 nth 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 .
© Walter de Gruyter Berlin · New York 2011
Articles in the same Issue
- Ricci soliton solvmanifolds
- Rational normal scrolls and the defining equations of Rees algebras
- Classifications of linear operators preserving elliptic, positive and non-negative polynomials
- Graded polynomial identities and exponential growth
- Un théorème de la masse positive pour le problème de Yamabe en dimension paire
- Degenerate problems with irregular obstacles
- Twisted cyclic theory, equivariant KK-theory and KMS states
- A trace formula for varieties over a discretely valued field
Articles in the same Issue
- Ricci soliton solvmanifolds
- Rational normal scrolls and the defining equations of Rees algebras
- Classifications of linear operators preserving elliptic, positive and non-negative polynomials
- Graded polynomial identities and exponential growth
- Un théorème de la masse positive pour le problème de Yamabe en dimension paire
- Degenerate problems with irregular obstacles
- Twisted cyclic theory, equivariant KK-theory and KMS states
- A trace formula for varieties over a discretely valued field
