Quadratic differentials in low genus: Exceptional and non-varying strata

Abstract

We give an algebraic way of distinguishing the components of the exceptional strata of quadratic differentials in genus three and four. The complete list of these strata is (9,-1), (6, 3,-1), (3, 3, 3,-1) in genus three and (12), (9, 3), (6, 6), (6, 3, 3) and (3, 3, 3, 3) in genus four. The upshot of our method is a detailed study regarding the geometry of canonical curves. This result is part of a more general investigation about the sum of Lyapunov exponents of Teich- m.ller curves, building on [9], [6] and [7]. Using disjointness of Teichm.ller curves with divisors of Brill- Noether type on the moduli space of curves, we show that for many strata of quadratic differentials in low genus the sum of Lyapunov exponents for the Teichm.ller geodesic flow is the same for all Teichm.ller curves in that stratum. © 2014 Société Mathématique de France. Tous droits réservés.