ABSTRACT
Contents 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 9.2 Linear Geometry Meets Commutative Algebra . . . . . . . . . . . . . . . . . . . . . 251
9.2.1 Arrangements with Few Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . 251 9.2.2 Ideals Generated by Products of Linear Forms . . . . . . . . . . . . . . 254
9.2.2.1 Coordinate Subspace Arrangements and the Taylor Complex . . . . . . . . . . . . . . . . . . . . . . . . . . 255
9.2.2.2 Examples of pl-Generated Arrangements . . . . . . . . . 256 9.2.2.3 Pl-Generated Arrangements and Resolutions . . . . . . 256 9.2.2.4 Subspace Arrangements That Are Not
pl-Generated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 9.3 The Castelnuovo-Mumford Regularity of Subspace Arrangements . . 258
9.3.1 A Regularity Bound via Interpolation . . . . . . . . . . . . . . . . . . . . . . . 260 9.3.2 Products of Linear Ideals and Resolutions . . . . . . . . . . . . . . . . . . 261
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
9.1 Introduction Let V be a vector space of dimension n + 1 over an infinite field k with
coordinate ring S = k[V ∗] ∼= k[x0, . . ., xn], and P(V ) = Pn be the projective space of lines through the origin in V . Let ˆX1, . . ., ˆXd be a collection of linear subspaces of V , among which there are no nontrivial containments, and Xi = P( ˆXi ). We call X = X1 ∪ · · · ∪ Xd a subspace arrangement in Pn.
