  In this paper, we study connected components of strata of the space of
quadratic differentials lying over $\T_g$. We use certain general properties of
sections of line bundles to put a upper bound on the number of connected
components, and a generalized version of the Gauss map as an invariant to put a
lower bound on the number of such components. For strata with sufficiently many
zeroes of the same order we can state precisely the number of components.
