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-müller curves, building on [9], [6] and [7]. Using disjointness of Teichmüller 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 Teichmüller geodesic flow is the same for all Teich-müller curves in that stratum.
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