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On the numerical rational connectedness of the Hilbert schemes of 2-points on rational surfaces

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Abstract

We prove that the Hilbert schemes of 2-points on rational surfaces are numerically rationally connected. The main idea is to show that certain 3-point genus-0 Gromov–Witten invariant of the Hilbert scheme of two points on the complex projective plane is positive and can be calculated enumeratively.

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Notes

  1. The terminology “symplectic rationally connected" is used in [27, 28]

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Acknowledgements

The authors would like to thank Professors Dan Edidin, Wei-Ping Li and Qi Zhang for stimulating discussions and valuable helps. The authors also would like to thank the referee for carefully reading the paper and providing valuable comments which have greatly improved the exposition of the paper.

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Authors and Affiliations

  1. School of Mathematics, Sun Yat-Sen University, 135 Xingang Road West, Guangzhou, China

    Jianxun Hu

  2. Department of Mathematics, University of Missouri, Columbia, MO, 65211, USA

    Zhenbo Qin

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  1. Jianxun Hu
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  2. Zhenbo Qin
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Correspondence to Zhenbo Qin.

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Jianxun Hu: Partially supported by National Natural Science Foundation of China (11831017, 11890662, 11771460, 11521101). Zhenbo Qin: Partially supported by a grant from the Simons Foundation.

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Hu, J., Qin, Z. On the numerical rational connectedness of the Hilbert schemes of 2-points on rational surfaces. manuscripta math. 162, 191–212 (2020). https://doi.org/10.1007/s00229-019-01122-z

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