A Derived Equivalence For A Del Pezzo Surface Of Degree 6 Over An Arbitrary Field

TL;DR

This paper establishes a derived category equivalence for degree six del Pezzo surfaces over any field, linking geometric and algebraic structures through an explicit finite-dimensional algebra.

Contribution

It proves a derived equivalence between the bounded derived category of coherent sheaves on the surface and modules over a specific algebra, extending previous classification and K-theory results.

Findings

Derived equivalence between geometric and algebraic categories

Explicit construction of the algebra A associated with the surface

Extension of classification results to arbitrary fields

Abstract

Let $S$ be a degree six del Pezzo surface over an arbitrary field $F$. Motivated by the first author's classification of all such $S$ up to isomorphism in terms of a separable $F$-algebra $B \times Q \times F$, and by his K-theory isomorphism $K_n(S) \cong K_n(B \times Q \times F)$ for $n \ge 0$, we prove an equivalence of derived categories $$ \sD^b(\coh S) \equiv \sD^b(\mod A) $$ where $A$ is an explicitly given finite dimensional $F$-algebra whose semisimple part is $B \times Q \times F$. Submitted to the Journal of K-theory