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

Abstract

AbstractThe Del Pezzo surface Y of degree 5 is the blow up of the plane in 4 general points, embedded in $${\mathbb {P}}^5$$ 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 $${\mathfrak {S}}_5$$ S 5 . We give canonical explicit $${\mathfrak {S}}_5$$ S 5 -invariant Pfaffian equations through a 6$$\times $$ × 6 antisymmetric matrix. We give concrete geometric descriptions of the irreducible representations of $${\mathfrak {S}}_5$$ S 5 . Finally, we give $${\mathfrak {S}}_5$$ S 5 -invariant equations for the embedding of Y inside $$({\mathbb {P}}^1)^5$$ ( P 1 ) 5 , and show that they have the same Hilbert resolution as for the Del Pezzo of degree 4.