On the Euler characteristics of certain moduli spaces of $1$-dimensional closed subschemes

Generalizing the ideas in [W. Li and Z. Qin, “On The Euler Number of Moduli Spaces of Curves and Points”, Commu. in Anal. and Geom. 14 (2006), 387–410] and using virtual Hodge polynomials as well as tours actions, we compute the Euler characteristics of certain moduli spaces of $1$‑dimensional closed subschemes when the ambient smooth projective variety admits a Zariski-locally trivial fibration to a codimension‑$1$ base. As a consequence, we partially verify a conjecture of W.‑P. Li and Qin. We also calculate the generating function for the number of certain punctual $3$‑dimensional partitions, which is used to compute the above Euler characteristics.

Keywords

partitions, moduli spaces of curves, Hilbert schemes, virtual Hodge polynomials, Donaldson–Thomas invariants, Euler characteristics

2010 Mathematics Subject Classification

Primary 14C05. Secondary 14D20, 14D21.

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Received 23 July 2020

Accepted 29 September 2020

Published 11 April 2021