Abstract
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.
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Altıntaş, A., Mond, D. Free resolutions for multiple point spaces. Geom Dedicata 162, 177–190 (2013). https://doi.org/10.1007/s10711-012-9722-4
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DOI: https://doi.org/10.1007/s10711-012-9722-4