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Geometric invariants for liaison of space curves. (English) Zbl 0596.14020
For a curve C in $${\mathbb{P}}^ 3$$ its Hartshorne-Rao module is $$M(C)=\oplus_{n\in {\mathbb{Z}}}M_ n(C),\quad where$$ $$M_ n(C)=H^ 1({\mathbb{P}}^ 1,{\mathcal I}_ C(n))$$. Upto duals and shifts, M(C) is known to be a complete liaison invariant of C. The ”degeneracy locus” of C is a collection $$\{V_ n\}$$, where $$V_ n$$ is a certain subvariety of the dual projective space $$({\mathbb{P}}^ 3)^*$$ defined in terms of the module structure action $$M_ n(C)\to M_{n+1}(C)$$ of $$H^ 0({\mathbb{P}}^ 3,{\mathcal O}(1))$$. Thus the degeneracy locus is a liaison invariant of C. In this paper the author investigates the relationship between the geometry of C and the degeneracy locus of C. More interesting is the application of his techniques and results to the determination of the liaison classes within a collection of certain types of curves in $${\mathbb{P}}^ 3$$, e.g., a union of skew lines or a general smooth rational curve, especially a quintic or a sextic. For a curve C the property of its being arithmetically C-M or Buchsbaum depends only upon M(C). So the techniques developed by the author lead him also to complete the answer to a question of Geramita, Maroscia and Vogel on the C-M and Buchsbaum properties of certain configurations of lines in $${\mathbb{P}}^ 3$$.
Reviewer: B.Singh

##### MSC:
 14H45 Special algebraic curves and curves of low genus 14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) 14N05 Projective techniques in algebraic geometry
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##### References:
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