Remarks on the castelnuovo-mumford regularity of subspace arrangements

Abstract

In this paper, I will survey a result of Harm Derksen and Jessica Sidman who proved that the ideal of an arrangement of d linear spaces in $\mathbb{P}^n$ is d-regular, answering a question posed by B.Sturmfels. For a finitely generated module M over a polynomial ring $S = k[x_0 , … , x_n]$ with k an algebraically closed field of arbitrary characteristic, we say that M is r-regular(in the sense of Castelnuovo and Mumford) if the $i^{th}$ syzygy module of $M$ is generated in degrees less than or equal to r + i. The notion of the regularity of M is key in determining the amount of computational resources that working with M requires. There is mysterious gap in our understanding of the behavior of the regularity of ideals. Namely, it is known that in the general case the regularity of an ideal may be as bad as doubly exponential in the degrees of its minimal generators and the number of variables in the ambient ring, and that this is essentially the worst case. However, for an ideal I corresponding to a smooth variety in characteristic zero, the regularity of I is linear in terms of geometric data ([1]). From this perspective, the ideals of subspace arrangements provide a test case for the behavior of the regularity of ideals of ideals of reduced but possibly singular schemes.