The combinatorics of Lehn's conjecture

Abstract

Let $S$ be a nonsingular projective surface equipped with a line bundle $H$. Lehn's conjecture is a formula for the top Segre class of the tautological bundle associated to $H$ on the Hilbert scheme of points of $S$. Voisin has recently reduced Lehn's conjecture to the vanishing of certain coefficients of special power series. The first result here is a proof of the vanishings required by Voisin by residue calculations (A. Szenes and M. Vergne have independently found the same proof). Our second result is an elementary solution of the parallel question for the top Segre class on the symmetric power of a nonsingular projective curve $C$ associated to a higher rank vector bundle $V$ on $C$. Finally, we propose a complete conjecture for the top Segre class on the Hilbert scheme of points of $S$ associated to a higher rank vector bundle on $S$ in the $K$-trivial case.