Abstract
Quot schemes of quotients of a trivial bundle of arbitrary rank on a nonsingular projective surface X carry perfect obstruction theories and virtual fundamental classes whenever the quotient sheaf has at most 1-dimensional support. The associated generating series of virtual Euler characteristics was conjectured to be a rational function in Oprea and Pandharipande (in, Geom Topol. http://arxiv.org/abs/1903.08787) when X is simply connected. We conjecture here the rationality of more general descendent series with insertions obtained from the Chern characters of the tautological sheaf. We prove the rationality of descendent series in Hilbert scheme cases for all curve classes and in Quot scheme cases when the curve class is 0.
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Notes
The conjecture can also be made for surfaces which are not simply connected, but we will not study non simply connected surfaces here (except in the \(\beta = 0 \) case).
The simplest geometry \(X={\mathbb {P}}^2\) places numerical restrictions leading, at least a priori, to less precise results regarding the denominators of the answers.
In the absence of (c), we have less control on the denominators of the rational functions thus obtained.
We have
$$\begin{aligned} 0\rightarrow {\mathcal {T}}{\mathcal {o}}{\mathcal {r}}^{1}_{{\mathcal {C}}\,} (\Omega _{{\mathcal {C}}/{\mathcal {B}}}, {\mathcal {O}}_{{\mathcal {C}}_b}) \rightarrow {\mathcal {N}}\big |_{{\mathcal {C}}_b}\rightarrow \Omega _{X}\big |_{{\mathcal {C}}_b}\rightarrow \Omega _{{\mathcal {C}}_b}\rightarrow 0. \end{aligned}$$\({\mathcal {T}}or^1\) is supported on the finitely many singularities of \({\mathcal {C}}_b\). Since \({\mathcal {N}}\big |_{{\mathcal {C}}_b}\) is locally free, \({\mathcal {T}}or^1\) vanishes. Therefore, \({\mathcal {N}}\big |_{{\mathcal {C}}_b}\) is the conormal bundle.
There are other roots which we will deal with later. See Eq. (29).
The overall q shift does not affect rationality.
This can also be seen via (5) since \(\text {Hilb}_{\beta }\) has negative virtual dimension.
We can show
$$\begin{aligned} {\mathsf {a}}=1-e_2t^2+2e_3t^3-3e_4t^4+\ldots \end{aligned}$$where \(e_i\) are the elementary symmetric functions in \(1, x_1, \ldots , x_{\ell }\). We do not explain the latter formula for \({\mathsf {a}}\) since it will not be used here.
The same calculation can also be carried out using Theorem 2.
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Acknowledgements
Our study of the virtual Euler characteristics of the Quot scheme of surfaces was motivated in part by the Euler characteristic calculations of L. Göttsche and M. Kool [12, 13] for the moduli spaces of rank 2 and 3 stable sheaves on surfaces. We thank A. Marian, W. Lim, A. Oblomkov, A. Okounkov, and R. Thomas for related discussions. D. J. was supported by SNF-200020-182181. D. O. was supported by the NSF through Grant DMS 1802228. R.P. was supported by the Swiss National Science Foundation and the European Research Council through Grants SNF-200020-182181, ERC-2017-AdG-786580-MACI, SwissMAP, and the Einstein Stiftung. We thank the Shanghai Center for Mathematical Science at Fudan University for a very productive visit in September 2018 at the start of the project. The project has received funding from the European Research Council (ERC) under the European Union Horizon 2020 Research and Innovation Program (Grant No. 786580).
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Johnson, D., Oprea, D. & Pandharipande, R. Rationality of descendent series for Hilbert and Quot schemes of surfaces. Sel. Math. New Ser. 27, 23 (2021). https://doi.org/10.1007/s00029-021-00638-1
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DOI: https://doi.org/10.1007/s00029-021-00638-1