The combinatorics of Lehn’s conjecture. (English) Zbl 1422.14008

For a rank-\(r\) vector bundle \(V\) on a smooth projective surface \(S\), there exists a tautological rank-\(rn\) bundle \(V^{[n]}\) on the Hilbert scheme \(S^{[n]}\) which parametrizes length-\(n\) \(0\)-dimensional closed subschemes of \(S\). In this paper, the authors study the top Segre class of the tautological bundle \(V^{[n]}\). When \(r = 1\) (i.e., \(V\) is a line bundle), a closed formula is obtained for the series \[ \sum_{n = 0}^\infty z^n \int_{S^{[n]}} s_{2n}(V^{[n]}), \] confirming a conjecture of M. Lehn [Invent. Math. 136, No. 1, 157–207 (1999; Zbl 0919.14001)]. The idea of the proof is to verify the vanishing of certain coefficients of special power series. Moreover, for an arbitrary rank \(r \ne 1\), the authors propose two conjectures regarding the series \[ \sum_{n = 0}^\infty z^n \int_{S^{[n]}} c_{2n}(V^{[n]}). \] Parallel results for the tautological bundles over the symmetric product of a smooth projective curve (viewed as the Hilbert schemes of points on the curve) are also proved.


14C05 Parametrization (Chow and Hilbert schemes)
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14Q05 Computational aspects of algebraic curves
14Q10 Computational aspects of algebraic surfaces


Zbl 0919.14001
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