Enumeration of singular subvarieties with tangency conditions

How many nodal degree d plane curves are tangent to a given line? The celebrated Caporaso-Harris recursion formula gives a complete answer for any number of nodes, degrees, and all possible tangency conditions. In this talk, I will report my work on the generalization of the counting problem to singular subvarieties with given tangency condition to a fixed smooth divisor on any smooth varieties. By establishing a beautiful correspondence with the algebraic cobordism group, we show the answer is given by universal polynomials.

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