${\mathfrak{S}}_5$-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 $6\times 6$ 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 $4$.

25 pages, more explicit formulae and pictures added. Final version to appear in the Rendiconti Circ. Mat. Palermo