Rank Zero Segre Integrals on Hilbert Schemes of Points on Surfaces

Abstract

The generating function of the Segre integrals on Hilbert schemes of points on a surface $X$ can be determined by five universal series $A_{0}(z)$, $A_{1}(z)$, $A_{2}(z)$, $A_{3}(z)$, $A_{4}(z)$, due to the result of Ellingsrud–Göttsche–Lehn. These five series do not depend on the surface $X$ and depend on the element $\alpha \in K(X)$, to which the Segre integrals are associated, only through the rank. Marian–Oprea–Pandharipande have determined $A_{0}(z),A_{1}(z),A_{2}(z)$ for all ranks. For rank 0, it is easy to see $A_{4}(z)=1$. Marian–Oprea–Pandharipande also conjectured that $A_{3}(z)=A_{0}(z)A_{1}(z)$ for rank 0. We prove this conjecture by showing that when $X$ is the projective plan, the Segre integrals associated to the structure sheaf of a curve in the anti-canoncial class are all zero. Hence, the rank zero Segre integrals on the Hilbert schemes of points for all surfaces are determined.