A note on the Severi problem for toric surfaces

In this note, we make a step towards the classification of toric surfaces admitting reducible Severi varieties. We generalize the results of [Lan19, Tyo13, Tyo14], and provide two families of toric surfaces admitting reducible Severi varieties. The first family is general, and is obtained by a quotient construction. The second family is exceptional, and corresponds to certain narrow polygons, which we call kites. We introduce two types of invariants that distinguish between the components of the Severi varieties, and allow us to provide lower bounds on the numbers of the components. The sharpness of the bounds is verified in some cases, and is expected to hold in general for ample enough linear systems. In the appendix, we establish a connection between the Severi problem and the topological classification of univariate polynomials.