Del Pezzo surfaces with a single $1/k(1,1)$ singularity

Abstract

Inspired by the recent progress by Coates-Corti-Kasprzyk et al. on Mirror Symmetry for del Pezzo surfaces, we show that for any positive integer k the deformation families of del Pezzo surfaces with a single 1 k (1, 1) singularity (and no other singular points) fit into a single cascade. Additionally we construct models and toric degenerations of these surfaces embedded in toric varieties in codimension ≤ 2. Several of these directly generalise constructions of Reid-Suzuki (in the case k = 3). We identify a root system in the Picard lattice, and in light of the work of Gross-Hacking-Keel, comment on Mirror Symmetry for each of these surfaces. Finally we classify all del Pezzo surfaces with certain combinations of 1 k (1, 1) singularities for k = 3, 5, 6 which admit a toric degeneration.DEL PEZZO SURFACES WITH A SINGLE 1 k (1, 1) SINGULARITY 3 du Val in the cases k = 1, 2, which are smooth and ordinary double points respectively. These are the only two cases for which the singularity 1 k (1, 1) is canonical. Definition 2.1. Given an arbitrary quotient singularity σ = 1 R (a, b), set k = gcd(a + b, R), c = (a + b)/k and r = R/k. Then σ can be written in the form 1 kr (1, kc − 1) and:Definition 2.1 is motivated by the work of Wahl [30] and Kollár-Shepherd-Barron [29] on the deformations of singularities. Discussion of these definitions from a toric viewpoint can be found in Akhtar-Kasprzyk [3]. A cyclic quotient singularity is a T -singularity if and only if it admits a Q-Gorenstein smoothing. Alternatively an R-singularity is rigid under any Q-Gorenstein deformation.Example 2.2. The singularities 1 k (1, 1) are R-singularities precisely when k = 3 or k ≥ 5. The singularities 1 2 (1, 1) and 1 4 (1, 1) are T -singularities. An algebraic surface is Q-Gorenstein if it is normal and the canonical divisor class is Q-Cartier.

Restrictions on the singularity content of a Fano polygon

“…Recent results support this conjecture [4,7,9], and understanding the possible values taken by the singularity content is an important open question. As a first step towards addressing this question, the two main results of this paper are:…”

Biregular models of log Del Pezzo surfaces with rigid singularities

“…It is called an R-singularity it it is is rigid under any Q-Gorenstein smoothing. We use the following characterization of a cyclic quotient singularity to be a T -singularity and R-singularity, appeared in [CP17].…”

“…Such surfaces with m × 1 3 (1, 1) points have been classified by Corti and Heuberger in [CH17]. The classification with a single orbifold point of type 1 r (1, 1) is provided by Cavey and Prince [CP17]. The log Del Pezzo surfaces have also been studied from the point of view of the existence of orbifold Kahler-Einstein metric on such surfaces in [CS13,CPS10,KP15] starting with [JK01].…”

“…The first model in the above table has already been discussed in [CP17] where its description as a toric variety has been provided.…”

“…The first model has been discussed in [CP17] and its toric description has also been provided by embedding it in a singular toric variety. In the case of Fano index 2 we get two biregular models which are indexed by 2 parameters giving rise to an infinite series of one parameter biregular models in each case.…”