A note on elliptic curves on toric surfaces

In this paper, we study the Severi varieties parametrizing integral curves of geometric genus-one on polarized toric surfaces in characteristic zero and describe their irreducible components. We show that the irreducible components are in natural bijection with certain affine sublattices of the lattice of characters of the toric surface. The sublattices are described explicitly in terms of the polygon defining the polarization of the toric surface.