Free resolutions for multiple point spaces

Abstract

This paper relates the multiple point spaces in the source and target of a corank 1 map-germ . Let f be such a map-germ, and, for 1 a parts per thousand currency sign k a parts per thousand currency sign multiplicity( f ), let D (k) ( f ) be its k'th multiple point scheme - the closure of the set of ordered k-tuples of pairwise distinct points sharing the same image. There are natural projections D (k+1)( f ) -> D (k) ( f ), determined by forgetting one member of the (k + 1)-tuple. We prove that the matrix of a presentation of over appears as a certain submatrix of the matrix of a suitable presentation of over . This does not happen for germs of corank > 1.