The irreducibility of the generalized Severi varieties

TL;DR

This paper proves that generalized Severi varieties, which parametrize certain plane curves with fixed tangency conditions, are irreducible using an inductive approach.

Contribution

It provides the first inductive proof of the irreducibility of generalized Severi varieties with specified tangency conditions.

Findings

Generalized Severi varieties are irreducible.

Inductive proof technique established for these varieties.

Confirms conjectures about their structure.

Abstract

We give an inductive proof that the generalized Severi varieties -- the varieties which parametrize (irreducible) plane curves of given degree and genus, with a fixed tangency profile to a given line at several general fixed points and several mobile points -- are irreducible.