I will present the current state of the art on the irreducibility problem for Severi
varieties on polarized toric surfaces. There exist several approaches to this problem.
In my talk, I will explore the tropical one, which applies in arbitrary
characteristic. In addition, I will provide examples and general constructions of
reducible Severi varieties and offer a complete classification of the irreducible
components in the genus-one case.
This talk is based on a series of joint works with Lionel Lang, Michael Barash, Karl
Christ, and Xiang He.
https://arxiv.org/abs/2212.00320
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