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Rational maps from products of curves to surfaces with \(p_g = q = 0\)

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Abstract

We study dominant rational maps from a product of two curves to surfaces with \(p_{g} = q = 0\). Given two curves which satisfy a mild genericity assumption and have large genus relative to their gonality, we show that the degree of irrationality of their product is equal to the product of their gonalities. Moreover, we prove that the degree of irrationality of a product of two hyperelliptic curves is 4.

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Notes

  1. By hyperelliptic curve, we mean a curve of genus at least 2 with a \(g^{1}_{2}\).

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Acknowledgements

We would like to thank Rob Lazarsfeld for valuable discussions and for suggesting that one could trace 2-forms to study low degree rational maps. We would also like to thank Mihnea Popa and Geoffrey Smith for helpful comments.

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Authors and Affiliations

  1. Department of Mathematics, Harvard University, Cambridge, MA, 02138, USA

    Nathan Chen

  2. Instituto Nacional de Matemática Pura e Aplicada, Rio de Janeiro, RJ, 22460-320, Brazil

    Olivier Martin

  3. Department of Mathematics, Stony Brook University, Stony Brook, NY, 11794, USA

    Olivier Martin

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  1. Nathan Chen
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Correspondence to Nathan Chen.

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Nathan Chen’s research was partially supported by an NSF postdoctoral fellowship, DMS-2103099.

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Chen, N., Martin, O. Rational maps from products of curves to surfaces with \(p_g = q = 0\). Math. Z. 304, 62 (2023). https://doi.org/10.1007/s00209-023-03312-8

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