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- Title
Projective curves of degree=codimension.
- Authors
Euisung Park
- Abstract
Abstract  In this article we study nondegenerate projective curves $${X \subset \mathbb{P}^{d-1}}$$ of degree d which are not arithmetically Cohen-Macaulay. Note that $${X = \pi_{P} (\widetilde{X})}$$ for a rational normal curve $${\widetilde{X} \subset \mathbb{P}^d}$$ and a point $${P \in \mathbb{P}^d \setminus \widetilde{X}^2}$$ . Our main result is about the relation between the geometric properties of X and the position of P with respect to $${\widetilde{X}}$$ . We show that the graded Betti numbers of X are uniquely determined by the rank $${\hbox{rk}_{\widetilde{X}} P}$$ of P with respect to $${\widetilde{X}}$$ . In particular, X satisfies property N 2,p if and only if $${p \leq \quad{rk}_{\widetilde{X}} P -3}$$ . Therefore property N 2,p of X is controlled by $${\quad{rk}_{\widetilde{X}} P}$$ and conversely $${\quad{rk}_{\widetilde{X}} P}$$ 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 141:1460â1478, 2005) holds. Also our result implies that for nondegenerate projective curves $${X \subset \mathbb{P}^{d-1}}$$ of degree d which are not arithmetically CohenâMacaulay, there are exactly $${\lfloor \frac{d-2}{2} \rfloor}$$ distinct Betti tables.
- Subjects
PROJECTIVE curves; MATHEMATICS; ALGEBRAIC curves; ALGEBRAIC varieties
- Publication
Mathematische Zeitschrift, 2007, Vol 256, Issue 3, p685
- ISSN
0025-5874
- Publication type
Article
- DOI
10.1007/s00209-007-0101-z