Asian Journal of Mathematics

Volume 21 (2017)

Number 3

Intersection numbers on the relative Hilbert schemes of points on surfaces

Pages: 531 – 542



Amin Gholampour (Department of Mathematics, University of Maryland, College Park, Md., U.S.A.)

Artan Sheshmani (Center of Mathematical Sciences and Applications (CMSA), Harvard University, Cambridge, Massachusetts, U.S.A.; and Department of Mathematics, Aarhus University, Aarhus, Denmark)


We study certain top intersection products on the Hilbert scheme of points on a nonsingular surface relative to an effective smooth divisor. We find a formula relating these numbers to the corresponding intersection numbers on the non-relative Hilbert schemes. In particular, we obtain a relative version of the explicit formula found by Carlsson–Okounkov for the Euler class of the twisted tangent bundle of the Hilbert schemes.


relative Hilbert schemes, intersection numbers, projective surface, Donaldson–Thomas invariants, modular forms

2010 Mathematics Subject Classification

14C05, 14C17, 14D21, 14F43

Received 31 July 2015

Published 5 July 2017