Geometric invariants for liaison of space curves
Geometric invariants for liaison of space curves
复制标题
空间曲线联络的几何不变量
DOI：
10.1016/0021-8693(86)90045-1
复制
发表时间：
1986
影响因子：
0.9
通讯作者：
J. Migliore
中科院分区：
文献类型：
--
作者：
J. Migliore
Let k be an algebraically closed field and let S= k [X,, X,, X2, X3]. For a curve Cc Pi, the Hartshorne-Rao module M (C)= BneiE H’(P3, Xc (n)) is a graded S-module of finite length and, up to duals and shifts, is a complete invariant of liaison (cf.[12 1). Given a graded S-module M=@,, z M, of finite length, the action of S,= P’(P’, 6 (1)) between two consecutive components (ie, Ql,,: S,+ Hom (M,, M,+,)) gives rise to a degeneracy locus, which can be thought of as lying in PSI=(P’)*. Namely, let I’,,,= P’(LES, Irk4,,(L)< r) and let V,= V,, s where s= max (r IV,., 5 (P3)*)(or else V,= 0). These loci are isomo~ hism invariants and are preserved under duals and shifts. For M= N (C) these loci are related to the geometry of C itself. The main philosophy that emerges (Section 2) is that the degeneracy locus generally corresponds to those planes H in P3 which meet C nongenerically, either containing a component of C or having Cn N impose an unusually small number of conditions on some plane curves on H. We conclude Section 2 by applying these ideas to derive some necessary conditions for M (C) to have components in negative degrees. The remaining sections give various apphcations of the techniques introduced in Section 2. First, in Section 3 we classify sets of skew lines up to liaison. That is, if C and C’consist of t 2 3 and t’skew lines, respectively, then we ask when C can be linked to C’. The answer in general is “never,” but the situation changes signi~ cantly if C lies on a quadric surface (cf. Theorem 3.1). A large part of the answer can be read from the degeneracy locus V,.
