On the projective embeddings of Gorenstein toric del Pezzo surfaces

Abstract

We study the projective embeddings of complete Gorenstein toric del Pezzo surfaces by ample complete linear systems, especially of minimal degree and dimension. Complete Gorenstein toric del Pezzo surfaces are in one-to-one correspondence with the 2-dimensional reflexive integral convex polytopes, which are classified into 16 types up to isomorphisms of lattices. Our main result shows that the minimal dimension and the minimal degree of all the ample complete linear systems on such a surface are attained by the primitive anti-canonical class except one case. From this, we determine the projective embeddings of these surfaces which are global complete intersections. We also show that the minimal free resolution of the defining ideal of the image under the anti-canonical embedding of these surfaces is given by an Eagon–Northcott complex.