ORBIFOLD POINTS ON PRYM-TEICHMULLER CURVES IN GENUS 4

摘要

For each discriminant D > 1, McMullen constructed the Prym-Teichmiiller curves W-D(4) and W-D(6) in M-3 and M-4, which constitute one of the few known infinite families of geometrically primitive Teichmiiller curves. In the present paper, we determine for each D the number and type of orbifold points on W-D (6). These results, together with a previous result of the two authors in the genus 3 case and with results of Lanneau-Nguyen and Moller, complete the topological characterisation of all Prym-Teichmiiller curves and determine their genus. The study of orbifold points relies on the analysis of intersections of W-D(6) with certain families of genus 4 curves with extra automorphisms. As a side product of this study, we give an explicit construction of such families and describe their Prym-Torelli images, which turn out to be isomorphic to certain products of elliptic curves. We also give a geometric description of the flat surfaces associated to these families and describe the asymptotics of the genus of W-D(6) for large D.