# $\mathfrak S_5$-equivariant syzygies for the Del Pezzo Surface of Degree 5

## TL;DR

This paper constructs explicit symmetric equations for a special algebraic surface, the degree 5 Del Pezzo surface, revealing its geometric and algebraic structure through group-invariant Pfaffian equations.

## Contribution

It provides canonical $rak S_5$-invariant Pfaffian equations and geometric descriptions for the Del Pezzo surface of degree 5, connecting its algebraic and symmetric properties.

## Key findings

- Explicit $rak S_5$-invariant Pfaffian equations derived.
- Geometric descriptions of $rak S_5$ irreducible representations.
- Equations for embedding into $(bP^1)^5$ with similar Hilbert resolution as degree 4 case.

## 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 $4$.