Title: Enumeration of singular varieties with tangency conditions,
Abstract: We construct the algebraic cobordism theory of bundles and divisors on
varieties. It has a simple basis (over Q) from projective spaces and its rank
is equal to the number of Chern numbers. An application of this algebraic
cobordism theory is the enumeration of singular subvarieties with give tangent
conditions with a fixed smooth divisor, where the subvariety is the zero locus
of a section of a vector bundle. We prove that the generating series of numbers
of such subvarieties gives a homomorphism from the algebraic cobordism group to
the power series ring. This implies that the enumeration of singular
subvarieties with tangency conditions is governed by universal polynomials of
Chern numbers, when the vector bundle is sufficiently ample. This result
combines and generalizes the Caporaso-Harris recursive formula, Gottsche's
conjecture, classical De Jonquiere's Formula and node polynomials from tropical
geometry.