### abstract

- Severi varieties were introduced by Severi [6] in 1920s in an attempt to prove the irreducibility of moduli spaces of algebraic curves of a given genus. He considered the case of plane curves and proved the irreducibility, but the proof contained a gap, and the first complete proof of the irreducibility was obtained by Harris [4] only in 1986. The result of Harris was generalized to other rational surfaces. In particular, the irreducibility is known for Hirzebruch surfaces [8], and in the case of rational curves it is also known for all toric surfaces [8] and all del Pezzo surfaces [7] with one trivial zero-dimensional exception: the variety of rational plane cubics through eight points in general position. The goal of this note is to describe an example of a toric surface admitting reducible positive-dimensional Severi varieties. The idea of Harris's proof of irreducibility of Severi varieties is as follows: A …