Rational Normal Scrolls and the Defining Equations of Rees Algebras

Details Rational Normal Scrolls and the Defining Equations of Rees Algebras Rational Normal Scrolls and the Defining Equations of Rees Algebras

2008-12-29

arxiv.org/abs/0812.4963v1

Mathematics

Commutative Algebra

48 pages

Scientific paper

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 $\f$ of $I$ has entries of degree $n$ and all of the other entries of $\f$ are linear. We identify an explicit generating set for the ideal $\Cal A$ which defines the Rees algebra $\Cal R=R[It]$; so $\Cal R=S/\Cal A$ for the polynomial ring $S=R[T_1,...,T_m]$. We resolve $\Cal R$ as an $S$-module and $I^s$ as an $R$-module, for all powers $s$. The proof uses the homogeneous coordinate ring, $A=S/H$, of a rational normal scroll, with $H\subseteq \Cal A$. The ideal $\Cal AA$ is isomorphic to the $n^{\text{th}}$ 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 $A/\Cal AA$ because the generators of $K^{(n)}$ are much less complicated then the generators of $\Cal AA$. 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 $\Cal C$ in projective $m-1$ space. The defining equations of the special fiber ring $\Cal R/(x,y)\Cal R$ yield a solution of the implicitization problem for $\Cal C$.

Affiliated with

Also associated with

No associations

Rating

Rational Normal Scrolls and the Defining Equations of Rees Algebras does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Rational Normal Scrolls and the Defining Equations of Rees Algebras, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Rational Normal Scrolls and the Defining Equations of Rees Algebras will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-117221