Projective curves of degree=codimension+2

Abstract

In this article we study nondegenerate projective curves X ⊂ P d−1 of degree d which are not arithmetically Cohen-Macaulay. Note that X = π P ( X) for a rational normal curve X ⊂ P d and a point P ∈ P d \ X 2 . Our main result is about the relation between the geometric properties of X and the position of P with respect to X. We show that the graded Betti numbers of X are uniquely determined by the rank rk X P of P with respect to X. In particular, X satisfies property N 2,p if and only if p ≤ rk X P − 3. Therefore property N 2,p of X is controlled by rk X P and conversely rk X P can be read off from the minimal free resolution of X. This result provides a non-linearly normal example for which the converse to Theorem 1.1 in (Eisenbud et al., Compositio Math 141:1460-1478, 2005 holds. Also our result implies that for nondegenerate projective curves X ⊂ P d−1 of degree d which are not arithmetically Cohen-Macaulay, there are exactly d−2 2 distinct Betti tables.

Smooth varieties of almost minimal degree

“…Property N p ,property N p and property N 2,p The Castelnuovo-Mumford regularity of X measures the number of nonzero rows of the Betti diagram of X. But in many cases, the first few steps of the minimal free resolution is much simpler than one can expect from reg(X).…”

On syzygies of non-complete embedding of projective varieties

“…In particular, the converse of Theorem 1.1 in [6] holds for π Q (X )(see Theorem 1.1 and Corollary 1.2 in [13]). For higher dimensional case, the problem is more complicated.…”

“…Recently the third author studied this problem when X ⊂ P r is a smooth rational normal scroll and t = 1 [13,14]. For example, let X ⊂ P r be a rational normal curve of degree r and let X k denote the k th higher secant variety of X .…”

Classification of Betti Diagrams of Varieties of Almost Minimal Degree

“…In this article we continue the study of Betti diagrams of varieties of almost minimal degree begun in [11] and [12].…”

“…Also when X is nonarithmetically Cohen-Macaulay, there have been several partial results concerned with the minimal free resolution of X (e.g. [1], [2], [7], [10], [11], [12]). In particular, M. Brodmann and P. Schenzel [1] gave a complete list of all occurring Betti diagrams in the case where codim(X, P r K ) ≤ 4.…”