The moving curve ideal and the Rees algebra

These are notes for a lecture given at Ohio University on June 3, 2006. An important topic in commutative algebra is the Rees algebra of an ideal in a commutative ring. The Rees algebra encodes a lot of information about the ideal and corresponds geometrically to a blow-up. One can represent the Rees algebra as the quotient of a polynomial ring by an ideal. This ideal is generated by the defining relations of the Rees algebra. The surprise is that these defining relations were discovered by computer scientists working in computer-aided geometric design, where the ideal is called the moving curve ideal. A second surprise will be the appearance of adjoint curves. 1. The Rees Algebra Let R be a Noetherian commutative ring. Given an ideal I ⊂ R and an R-module M , we can build three graded rings: • The Rees algebra R(I) = R⊕ I ⊕ I ⊕ · · · • The associated graded grI(R) = R/I ⊕ I/I 2 ⊕ I/I ⊕ · · · • The symmetric algebra SymR(M) = R⊕M ⊕ Sym (M)⊕ · · · These are related as follows: grI(R) = R(I)⊗R R/I. There are natural surjections SymR(I) Φ −→ R(I) SymR/I(I/I ) χ −→ grI(R) Proj(R(I)) is the blow-up of V(I) ⊂ Spec(R). Here are two basic results: Theorem Φ is an isomorphism ⇐⇒ χ is an isomorphism. Theorem If R is local and grI(R) is Cohen-Macaulay, then R(I) is Cohen-Macaulay. Example 1.1 R is a regular local ring and I ⊂ R is generated by a regular sequence f1, . . . , fr. Then one can prove that we have a 1