Irena Peeva

S OF SOME OF MY PAPERS GROUPED BY TOPIC 1. Regularity. Let k be an algebraically closed field. We will work over a standard graded polynomial ring U over k. Projective dimension and regularity are the main numerical invariants that measure the complexity of a minimal free resolution. Let L be a homogeneous ideal in U , and let b ij (L) be its graded Betti numbers. The projective dimension pd(L) = max i b i,j (L) 6= 0 is the index of the last non-zero column of the Betti table b(L) := b i,i+j(L) , and thus it measures its width. The height of the table is measured by the index of the last non-zero row and is called the (Castelnuovo-Mumford) regularity of L; it is defined as reg(L) = max j b i, i+j(L) 6= 0 . Alternatively, regularity can be defined using local cohomology. See the book [Ei3] for a detailed discussion on regularity. Hilbert’s Syzygy Theorem provides a nice upper bound on the projective dimension: pd(L) is smaller than the number of variables. In contrast, Giusti and Galligo (see [BM, Theorem 3.7]), and CavigliaSbarra [CS] proved a doubly exponential upper bound on the regularity of homogeneous ideals. The bound is in terms of the number of variables and the degrees of the minimal generators of L. It is nearly sharp since the Mayr-Meyer construction [MM] leads to examples of families of ideals attaining doubly exponential regularity; such examples were constructed by Bayer-Mumford [BM], Bayer-Stillman [BS], and Koh [Ko]. It is expected that much better bounds hold for the defining ideals of geometrically nice projective varieties. In the smooth case, important bounds were obtained by Mumford [BM, Theorem 3.12(b)], Bertram-Ein-Lazarsfeld [BEL], and Chardin-Ulrich [CU]. As discussed in the influential paper [BM] by Bayer and Mumford, “the biggest missing link” between the general case and the smooth case is to obtain a “decent bound on the regularity of all reduced equidimensional ideals”. The longstanding Regularity Conjecture predicts the following elegant linear bound in terms of the degree: Regularity Conjecture 1.1. (Eisenbud-Goto, 1984) [EG] Suppose that the field k is algebraically closed. If L ⇢ (z1, . . . , zp) is a homogeneous prime ideal in U = k[z1, . . . , zp], then reg(L)  deg(U/L) codim(L) + 1 , where deg(U/L) is the degree of U/L (also called the multiplicity of U/L), and codim(L) is the codimension (also called height) of L. The condition that L ⇢ (z1, . . . , zp) is equivalent to requiring that the projective variety V (L) is not contained in a hyperplane in Pp 1 k . Prime ideals that satisfy this condition are called non-degenerate. The Regularity Conjecture holds if U/L is Cohen-Macaulay by a result of Eisenbud-Goto [EG]. It is proved for curves by Gruson-Lazarsfeld-Peskine [GLP], completing classical work of Castelnuovo. It is also proved for smooth surfaces by Lazarsfeld [La] and Pinkham [Pi], and for most smooth 3-folds by Ran [Ra]. In the smooth case, Kwak ([Kw] gave bounds for regularity in dimensions 3 and 4 that are only slightly worse than the optimal ones. Many other special cases and related bounds have been proved as well.