Degeneration of curves on some polarized toric surfaces

Abstract We address the following question: Given a polarized toric surface ( S mathvariant="normal">, L stretchy="false">) {(S,{\mathcal{L}})} , and general integral curve C of geometric genus g in linear system stretchy="false">| {|{\mathcal{L}}|} do there exist degenerations to curves smaller genera? give an affirmative answer this question for surfaces associated h -transverse polygons, provided that characteristic ground field is large enough. examples small characteristic, which negative. In case affirmative, we deduce as above nodal. 0, use result show irreducibility Severi varieties class with polygon.