Notes on liaison and duality
Peter Schenzel · Kyoto journal of mathematics · 1982
In classifying algebraic curves in projective three space N o e th e r [8 ] introduced th e n o tio n o f th e residual intersection (liaison).Two curves C, k a n algebraically closed field, are linked geometrically, i f th ey h a v e n o components i n common a n d C U C ' i s a complete intersection.In his paper [10] Rao studied t h e following graded S-m odule : M(C)--= (1) H l(P , 2 c (s)), whereand S c denotes the ideal sheaf of C .If C, C' a re linked, then M(C) and M(C') are isomorphic up to duality and shifts in g ra d in g s.One of the m ain results o f our paper is to extend R ao's result to a n arbitrary Cohen-Macaulay variety X O E P .That is, we define a formal vector m (X ) which is shown to be an invariant up to duality and shifts in gradings under the liaison, compare 5.