A comb of origami curves in the moduli space M₃ with three dimensional closure

The first part of this paper is a survey on Teichmueller curves and Veech groups, with emphasis on the special case of origamis where much stronger tools for the investigation are available than in the general case. In the second part we study a particular configuration of origami curves in genus 3: A "base" curve is intersected by infinitely many "transversal" curves. We determine their Veech groups and the closure of their locus in M_3, which turns out to be a three dimensional variety and the image of a certain Hurwitz space in M_3.