  Poisson structures of divisor-type are those whose degeneracy can be captured
by a divisor ideal, which is a locally principal ideal sheaf with nowhere-dense
quotient support. This is a large class of Poisson structures which includes
all generically-nondegenerate Poisson structures, such as log-, $b^k$-,
elliptic, elliptic-log, and scattering Poisson structures. Divisor ideals are
used to define almost-injective Lie algebroids of derivations preserving them,
to which these Poisson structures can be lifted, often nondegenerately so. The
resulting symplectic Lie algebroids can be studied using tools from symplectic
geometry. In this paper we develop an effective framework for the study of
Poisson structures of divisor-type (also called almost-regular Poisson
structures) and their Lie algebroids. We provide lifting criteria for such
Poisson structures, develop the language of divisors on smooth manifolds, and
discuss residue maps to extract information from Lie algebroid forms along
their degeneraci loci.
