  This paper lays the foundation for determining the Kodaira dimension of the
projectivized strata of Abelian differentials with prescribed zero and pole
orders in large genus. We work with the moduli space of multi-scale
differentials constructed in [BCGGM2] which provides an orbifold
compactification of these strata. We establish the projectivity of the moduli
space of multi-scale differentials, describe the locus of canonical
singularities, and compute a series of effective divisor classes. Moreover, we
exhibit a perturbation of the canonical class which allows the corresponding
pluri-canonical forms to extend over the locus of non-canonical singularities.
  As applications, we certify general type for strata with few zeros as well as
for strata with equidistributed zero orders when g is sufficiently large. In
particular, we show general type for the odd spin components of the minimal
strata for g > 12.
