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. This result is part of a more general investigation of disjointness of Teichmuller curves with divisors of Brill-Noether type on the moduli space of curves. As a result we show that for many strata of quadratic differentials in low genus the sum of Lyapunov exponents for the Teichmuller geodesic flow is the same for all Teichmuller curves in that stratum.