LDP Polygons and the Number 12 Revisited

Abstract

We give a combinatorial proof of a lattice point identity involving a lattice polygon and its dual, generalizing the formula $\operatorname{area}\left(\Delta \right) + \operatorname{area}\left(\Delta^* \right) = 6$ for reflexive $\Delta$. The identity is equivalent to the stringy Libgober-Wood identity for toric log del Pezzo surfaces.