Abstract
We provide explicit graded constructions of orbifold del Pezzo surfaces with rigid orbifold points of type ki×1ri(1,ai):3≤ri≤10,ki∈Z≥0 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 5×5 skew symmetric matrix, and nine 2×2 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 MAGMA.
Highlights
A del Pezzo surface is a two dimensional algebraic variety with an ample anti-canonical divisor class
We describe X to be locally qGorenstein(qG)-rigid if it contains only rigid isolated orbifold points, i.e., the orbifold points are rigid under qG-deformations
In cases of hypersurfaces and complete intersections, the classifications of tuples (d j ; ai ) which give rise to a quasismooth del Pezzo surfaces can be found in [17,18] where d j denote the degrees of the defining equations and ai are weights of the ambient weighted projective space
Summary
A del Pezzo surface is a two dimensional algebraic variety with an ample anti-canonical divisor class. We describe X to be locally qGorenstein(qG)-rigid if it contains only rigid isolated orbifold points, i.e., the orbifold points are rigid under qG-deformations If it admits a qG-degeneration to a normal toric del Pezzo surface it is called a del Pezzo surface of class TG. The classification of orbifold del Pezzo surfaces has received much attention, primarily due to the mirror symmetry program for Fano varieties by Coates, Corti et al [3]. The mirror symmetry for orbifold del Pezzo surface has been formulated in [4] in the form of a conjecture expecting a one to one correspondence between mutation equivalence classes of Fano polygons with the (qG)-deformation equivalence classes of locally qG-rigid del Pezzo surfaces of class TG. Gave the classification of locally qG-rigid del Pezzo surfaces with 13 (1, 1) singular points. Pezzo surfaces with certain basket of singularities which can be described by relatively small sets of equations
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