In this article we study nondegenerate projective curves of degree d which are not arithmetically Cohen-Macaulay. Note that for a rational normal curve and a point. Our main result is about the relation between the geometric properties of X and the position of P with respect to We show that the graded Betti numbers of X are uniquely determined by the rank of P with respect to. In particular, X satisfies property N 2,p if and only if. Therefore property N 2,p of X is controlled by and conversely 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 of degree d which are not arithmetically Cohen-Macaulay, there are exactly distinct Betti tables.
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