Polarized rigid del Pezzo surfaces in low codimension

Abstract

We provide explicit graded constructions of orbifold del Pezzo surfaces with rigid orbifold points of type $\left\{k_i\times\frac{1}{r_i}(1,a_i): 3\le r_i \le 10,k_i \in \ZZ_{\ge 0}\right\}$; as well-formed and quasismooth varieties embedded in some weighted projective space. In particular, we present a collection of 147 such surfaces such that their image under their anti-canonical embeddings can be described by using one of the following sets of equations: a single equation, two linearly independent equations, five maximal Pfaffians of $5\times 5$ skew symmetric matrix, and nine $2\times 2$ minors of size 3 square matrix. This is a complete classification of such surfaces under certain carefully chosen bounds on the weights of ambient weighted projective spaces and it is largely based on detailed computer-assisted searches by using the computer algebra system \textsc{magma}.

22 pages, 5 tables, An improved version after a referee's comments

References in corpus (3)

• Weighted complete intersection del Pezzo surfaces
• Classification of del Pezzo surfaces with $\frac{1}{3}(1,1)$- and $\frac{1}{4}(1,1)$-singularities
• Classification of Minimal Polygons with Specified Singularity Content