Enumeration of singular varieties with tangency conditions
We construct the algebraic cobordism theory of bundles and divisors on smooth varieties. It has a simple basis (over Q) from projective spaces and its rank is equal to the number of Chern invariants. As an application we study the number of singular subvarieties satisfying given 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 gives a homomorphism from the algebraic cobordism group to the power series ring. This implies that the number 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 and Vakil's recursive formulae, Gottsche's conjecture, De Jonquiere's Formula and relative node polynomials from tropical geometry. In the revised version, we allow the tangency conditions to be at assigned and unassigned points and prove the existence of universal polynomials and a formula for generating series. This completes the most general case of such enumeration of singular varieties for any dimension. We also correct the proof in Section 5, 6 when the divisor D is not irreducible, and Theorem 2.2. Typos are corrected.