A note on the Severi problem for toric surfaces

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