Bifurcation set of multi-parameter families of complex curves

摘要

The problem of detecting the bifurcation set of polynomial mappings CmCk, m2, mk1, has been solved in the case m=2, k=1 only. Its solution, which goes back to the 1970s, involves the non-constancy of the Euler characteristic of fibers. We provide here a complete answer to the general case m=k+13 in terms of the Betti numbers of fibers and of a vanishing phenomenon discovered in the late 1990s in the real setting.