Free resolutions for multiple point spaces

This paper relates the multiple point spaces in the source and target of a corank 1 map-germ ${(mathbb {C}^n, 0)to(mathbb {C}^{n+1}, 0)}$ . Let f be such a map-germ, and, for 1 ≤ k ≤ 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 ${mathcal {O}_{D^{k+1}(f)}}$ over ${mathcal {O}_{D^k(f)}}$ appears as a certain submatrix of the matrix of a suitable presentation of ${mathcal {O}_{mathbb {C}^n,0}}$ over ${mathcal {O}_{mathbb {C}^{n+1},0}}$ . This does not happen for germs of corank > 1.