Reduction numbers, Rees algebras, and Pfaffian ideals

I. Aberbach, S. Huckaba, C. Huneke

Assuming that (R,m) is a Cohen-Macaulay local ring with infinite residue field and I is an ideal of R having analytic deviation 2, we provide a condition (in terms of a presentation matrix of I, and inspired by work of Vasconcelos) that forces bounds on the reduction number of I. We proceed to apply the condition to various situations. Our main application is to a certain family of 5-generated height 3 Gorenstein ideals of a regular local ring. This application is possible by making use of the structure theorem of Buchsbaum and Eisenbud to express these Gorenstein ideals in terms of the Pfaffians of a 5 x 5 skew-symmetric presentation matrix of I. The applications help to produce Cohen-Macaulay Rees algebra results.