Abstract
In this paper, we show that Severi varieties parameterizing irreducible reduced planar curves of a given degree and geometric genus are either empty or irreducible in any characteristic. Following Severi’s original idea, this gives a new proof of the irreducibility of the moduli space of smooth projective curves of a given genus in positive characteristic. It is the first proof that involves no reduction to the characteristic zero case. As a further consequence, we generalize Zariski’s theorem to positive characteristic and show that a general reduced planar curve of a given geometric genus is nodal.
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21 December 2022
The original online version of this article was revised. The Article History was given incorrectly, and missing References added.
20 December 2022
A Correction to this paper has been published: https://doi.org/10.1007/s10240-022-00137-9
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IT is partially supported by the Israel Science Foundation (grant No. 821/16). KC is supported by the Israel Science Foundation (grant No. 821/16) and by the Center for Advanced Studies at BGU. XH was supported by the ERC Consolidator Grant 770922 – BirNonArchGeom.
The original online version of this article was revised. The Article History was given incorrectly, and missing References added.
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Christ, K., He, X. & Tyomkin, I. On the Severi problem in arbitrary characteristic. Publ.math.IHES 137, 1–45 (2023). https://doi.org/10.1007/s10240-022-00135-x
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DOI: https://doi.org/10.1007/s10240-022-00135-x