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Hyperelliptic odd coverings

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Abstract

We investigate a class of odd (ramification) coverings C → ℙ1 where C is hyperelliptic, its Weierstrass points map to one fixed point of ℙ1 and the covering map makes the hyperelliptic involution of C commute with an involution of ℙ1. We show that the total number of hyperelliptic odd coverings of minimal degree 4g is \(\left({\matrix{{3g} \cr {g - 1} \cr}} \right){2^{2g}}\) when C is general. Our study is approached from three main perspectives: if a fixed effective theta characteristic is fixed they are described as a solution of a certain class of differential equations; then they are studied from the monodromy viewpoint and a deformation argument that leads to the final computation.

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Authors and Affiliations

  1. Dipartimento di Matematica “F. Casorati”, Università degli Studidi Pavia, Via Ferrata 5, 27100, Pavia, Italy

    Riccardo Moschetti & Gian Pietro Pirola

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  1. Riccardo Moschetti
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  2. Gian Pietro Pirola
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Correspondence to Riccardo Moschetti.

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Moschetti, R., Pirola, G.P. Hyperelliptic odd coverings. Isr. J. Math. 249, 477–500 (2022). https://doi.org/10.1007/s11856-022-2318-2

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  • DOI: https://doi.org/10.1007/s11856-022-2318-2

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