Multiple point spaces of finite holomorphic maps

作者：Nuno Ballesteros Juan J^*; Penafort Sanchis Guillermo

来源：The Quarterly Journal of Mathematics, 2017, 68(2): 369-390.

DOI：10.1093/qmath/haw042

Given a finite holomorphic map f : X -> Y between manifolds, we show that there exists a unique possible definition of kth multiple point space D-k(f) with the following properties: Dk(f) is a closed subspace of X-k, D-k(f) is the closure of the set of strict k-multiple points when f is stable and D-k(f) is well behaved under deformations. Our construction coincides with Mond's double point space and Mond's kth multiple point space for corank one singularities. We also give some interesting properties of the double point space and prove that in many cases it can be computed as the zero locus of the quotient of ideals (f x f)* (phi)Delta(Y) : (phi)Delta(X), where (phi)Delta(Z) is the defining ideal sheaf of the diagonal in Z x Z.

• 出版日期2017-6