Commutative Algebra and Algebraic Geometry (Eisenbud Seminar)
Jason McCullough (Iowa State University)
Fix a polynomial ring R = K[x_1...x_n] and a homogeneous ideal I in R. Let d denote the maximal degree of generator of I. A result of Galligo, Giusti and Caviglia-Sbarra shows that reg(R/I) is at most doubly exponential in terms of d and n. Examples due to Mayer and Mayr show that any upper bound must be doubly exponential. However, all extremal examples of resolutions that have large regularity seem to have large regularity early in the resolution. So it makes sense that better upper bounds should be possible if one uses more data about the resolution than just d and n. In this talk, I\'ll show how one can prove two such bounds using the numerics of the Boij-Soederberg decomposition of the Betti table of R/I.