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# Asian Journal of Mathematics

## Volume 21 (2017)

### Number 3

### Intersection numbers on the relative Hilbert schemes of points on surfaces

Pages: 531 – 542

DOI: https://dx.doi.org/10.4310/AJM.2017.v21.n3.a6

#### Authors

#### Abstract

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.

#### Keywords

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