Reducible Families of Curves with Ordinary Multiple Points on Surfaces in IP3 C

For a Coxeter system (G, S) the multi-parametric alternating subalgebra H ⁺(G) of the Hecke algebra and the alternating subgroup ?⁺(G) of the braid group are defined. Two presentations for H ⁺(G) and ?⁺(G) are given; one generalizes the Bourbaki presentation for the alternating subgroups of Coxeter groups, another one uses generators related to edges of the Coxeter graph.