Multiple point spaces of finite holomorphic maps

2015

We show that there exists a unique possible definition, with certain natural properties, of the multiple point space of a holomorphic map between complex manifolds. Our construction coincides with the double point space and the k-th multiple point space for corank one map-germs, due to Mond. 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 certain quotient of ideals.

Double point curves for corank 2 map germs from C2 to C3

AbstractWe characterize finite determinacy of map germs f:(C2,0)→(C3,0) in terms of the Milnor number μ(D(f)) of the double point curve D(f) in (C2,0) and we provide an explicit description of the double point scheme in terms of elementary symmetric functions. Also we prove that the Whitney equisingularity of 1-parameter families of map germs ft:(C2,0)→(C3,0) is equivalent to the constancy of both μ(D(ft)) and μ(ft(C2)∩H) with respect to t, where H⊂C3 is a generic plane.

Double point curves for corank 2 map germs from C2 to C3

2012

Abstract We characterize finite determinacy of map germs f : ( C 2 , 0 ) → ( C 3 , 0 ) in terms of the Milnor number μ ( D ( f ) ) of the double point curve D ( f ) in ( C 2 , 0 ) and we provide an explicit description of the double point scheme in terms of elementary symmetric functions. Also we prove that the Whitney equisingularity of 1-parameter families of map germs f t : ( C 2 , 0 ) → ( C 3 , 0 ) is equivalent to the constancy of both μ ( D ( f t ) ) and μ ( f t ( C 2 ) ∩ H ) with respect to t , where H ⊂ C 3 is a generic plane.