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arXiv · 2005.04134

On the Severi problem in arbitrary characteristic

Karl Christ·Xiang He·Ilya Tyomkin
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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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Karl Christ, Xiang He, Ilya Tyomkin. 2020-05-08. On the Severi problem in arbitrary characteristic. https://doi.org/10.1007/s10240-022-00135-x

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