Enumeration of singular curves with tangency conditions
Enumeration of singular curves with tangency conditions

YuJong Tzeng , Harvard University
Fine Hall 322
How many nodal degree d plane curves are tangent to a given line? The celebrated CaporasoHarris recursion formula gives a complete answer for any number of nodes, degrees, and all possible tangency conditions. In this talk, I will report my recent work on the generalization of the above problem to count singular curves with given tangency condition to a fixed smooth divisor on general surfaces. I will relate the enumeration to tautological integrals on Hilbert schemes of points and show the numbers of curves in question are given by universal polynomials. As a result, we can obtain infinitely many new formulas for nodal curves and understand the asymptotic behavior for all singular curves with any tangency conditions.