Mapping class groups of trigonal loci

Abstract

In this paper, we study the topology of the stack $$\mathcal {T}_g$$ of smooth trigonal curves of genus g over the complex field. We make use of a construction by the first named author and Vistoli, which describes $$\mathcal {T}_g$$ as a quotient stack of the complement of the discriminant. This allows us to use techniques developed by the second named author to give presentations of the orbifold fundamental group of $$\mathcal {T}_g$$ , and of its substrata with prescribed Maroni invariant, and describe their relation with the mapping class group $$\mathcal {M}ap_g$$ of Riemann surfaces of genus g.