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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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