On the projective embeddings of Gorenstein toric del Pezzo surfaces (Q610638)

The authors study the embeddings of complete Gorenstein toric del Pezzo (GTDP) surfaces into projective spaces given by ample complete linear systems, especially of minimal dimension and minimal degree. A projective algebraic variety \(X\) of dimension \(2\) is called Gorenstein del Pezzo variety if the anti-canonical divisor \(-K_X\) is an ample Cartier divisor. A classification of Gorenstein del Pezzo surfaces is known due to \textit{M. Demazure} [Lect. Notes Math. 777, 21--69 (1980; Zbl 0444.14024)]. The GTDP varieties of dimension \(n\) are uniquely determined up to isomorphism of lattice polytopes by the corresponding reflexive lattice polytope. In the case \(n=2\) the classification of reflexive lattice polytopes is well known [cf. \textit{Batyrev}, Higher dimensional toric varieties with ample anticanonical class, Moskow (Russian), PhD. Thesis (1985); \textit{S. Rabinowitz}, Ars Comb. 28, 83--96 (1989; Zbl 0704.52006)]. There exist exactly 16 non-isomorphic reflexive lattice convex polygons that are shown in Figure 1 [cf. \textit{R. J. Koelman}, The number of moduli of families of curves on toric surfaces, PhD. Thesis, Nijmegen (1991); \textit{B. Poonen} and \textit{F. Rodriguez-Villegas}, Am. Math. Mon. 107, No. 3, 238--250 (2000; Zbl 0988.52024)]. The one-to-one correspondence between GTDP surfaces with the 2-dimensional reflexive lattice convex polytopes is given via the classical construction of toric varieties from the fan associated to a polytope as usual in toric geometry. In Theorem 2.1 they show that the projective embedding of minimal dimension and minimal degree by ample complete linear systems on GTDP surfaces is given simultaneously by the primitive anti-canonical class except for the one that corresponds to the blowing-up of \(\mathbb P^2\) at a point. For the proof the authors proceed by analyzing each case. In the paper they only show one case that they study in great detail. Since these Gorenstein surfaces have at worst \(A_n\)-singularities, they are actually locally complete intersection. In Theorem 2.2 they determine which of the GTDP surfaces are globally complete intersections in projective spaces. Finally, they consider the defining ideal \(I\) of the image of GTDP surfaces under the anti-canonical embedding as subvarieties of \(\mathbb P^N\), and the syzygies of \(I\). In Proposition 2.4 they show that the minimal free resolution of \(I\) is given by an Eagon-Northcott complex. This paper is based on the master's thesis of the first author.