{mathfrak {S}}_5-equivariant syzygies for the Del Pezzo surface of degree 5

Abstract

The Del Pezzo surface Y of degree 5 is the blow up of the plane in 4 general points, embedded in {mathbb {P}}^5 by the system of cubics passing through these points. It is the simplest example of the Buchsbaum–Eisenbud theorem on arithmetically-Gorenstein subvarieties of codimension 3 being Pfaffian. Its automorphism group is the symmetric group {mathfrak {S}}_5. We give canonical explicit {mathfrak {S}}_5-invariant Pfaffian equations through a 6times 6 antisymmetric matrix. We give concrete geometric descriptions of the irreducible representations of {mathfrak {S}}_5. Finally, we give {mathfrak {S}}_5-invariant equations for the embedding of Y inside ({mathbb {P}}^1)^5, and show that they have the same Hilbert resolution as for the Del Pezzo of degree 4.