Classification of log del Pezzo surfaces of index three

Abstract

Abstract. A normal projective non-Gorenstein log-terminal surface S is called a log del Pezzo surface of index three if the threetimes of the anti-canonical divisor −3K S is an ample Cartier divisor. We classify all of the log del Pezzo surfaces of index three. The technique for the classification based on the argument of Nakayama.

Del Pezzo surfaces with $$\frac{1}{3}(1,1)$$ 1 3 ( 1 , 1 ) points

“…We further structure the classification into six unprojection cascades, determine their biregular invariants and their directed MMP together with a distinguished configuration of negative curves on the minimal resolution. This overlaps with work of Fujita and Yasutake [14].…”

“…We further structure the classification into six unprojection cascades, determine their biregular invariants and their directed MMP together with a distinguished configuration of negative curves on the minimal resolution. This overlaps with work of Fujita and Yasutake [14].(II) Construct good models for surfaces in all families as degeneracy loci in rep quotient varieties. In all but two cases, the rep quotient variety is a simplicial toric variety.(III) Prove that precisely 26 of the 29 families admit a qG-degeneration to a toric surface.The classification is summarised in table 1 and table 2, which also plot invariants and provide good model constructions of surfaces in all families.This work is part of a program to understand mirror symmetry for orbifold del Pezzo surfaces [2,16,26,31,27,10] and it is aimed specifically at giving evidence for the conjectures made in [2].The rest of the introduction is organised as follows: in § 1.1 we give precise statements of our main results; in § 1.2 we say a few words about the context of [2]; in § 1.3 we outline the structure of the paper.…”

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We classify non-smooth del Pezzo surfaces with 1 3 (1, 1) points in 29 qG-deformation families grouped into six unprojection cascades (this overlaps with work of Fujita and Yasutake [14]), we tabulate their biregular invariants, we give good model constructions for surfaces in all families as degeneracy loci in rep quotient varieties, and we prove that precisely 26 families admit qG-degenerations to toric surfaces. This work is part of a program to study mirror symmetry for orbifold del Pezzo surfaces [2].In this paper we:(I) Classify non-smooth del Pezzo surfaces with 1 3 (1, 1) points in precisely 29 qG-deformation families.

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Toric log del Pezzo surfaces with one singularity

“…Related results are due to Kojima [31] (whenever the Picard number equals 1) and Nakayama [33] (whose techniques apply even if one replaces C with an algebraically closed field of arbitrary characteristic). Based on Nakayama's arguments, Fujita & Yasutake [22] succeeded recently to extend the classification even for = 3. But for indices ≥ 4 the situation turns out to be much more complicated, and (apart from some partial results as those in [20], [21]) it is hard to expect a complete characterization of these surfaces in this degree of generality.…”

Smoothing toric Fano surfaces using the Gross-Siebert algorithm

“…There has been much recent interest in the classification of log del Pezzo surfaces up to deformation — in particular log del Pezzo surfaces have been classified in index at most two by Alexeev–Nikulin and in index three by Fujita–Yasutake . Here we analyse ‐Gorenstein deformations of del Pezzo surfaces with cyclic quotient singularities, exploring a rich combinatorial structure predicted to exist by mirror symmetry.…”