Semiorthogonal decompositions and birational geometry of del Pezzo surfaces over arbitrary fields

Abstract

We study the birational properties of geometrically rational surfaces from a derived categorical perspective. In particular, we give a criterion for the rationality of a del Pezzo surface S over an arbitrary field, namely, that its derived category decomposes into zero‐dimensional components. When S has degree at least 5 we construct explicit semiorthogonal decompositions by subcategories of modules over semisimple algebras arising as endomorphism algebras of vector bundles and we show how to retrieve information about the index of S from Brauer classes and Chern classes associated to these vector bundles.

Derived categories of quintic del Pezzo fibrations

“…Then f is injective and X = Gr(2, 5) ∩ P 5 ⊂ P 9 . By symmetry of Gr(2, 5) ∼ = Gr (3,5), the same result holds if N is of rank 3.…”

“…the Azumaya algebras B i that appear will be trivial. It is true, for example, when X is a smooth del Pezzo surface over a field of degree at least 5, see [3]. Because a quintic del Pezzo surface with rational Gorenstein singularities over a field is always rational (see [30] for the smooth case and [8, Theorem 9.1(b)] for the singular case), one could anticipate that no nontrivial Azumaya algebras would appear in the semiorthogonal decomposition of the quintic del Pezzo fibration and the main result of the paper confirms the anticipation.…”

“…: From direct computations, we have RHom(Sym 3 R * , O(2)) = 0 and RHom(Sym3 R * , (Sym 2 R * )(1)) = H 0 (Gr(2, V ), R * ) = V * . Right mutations of the triple (Sym 3 R * , O(2), (Sym 2 R * )(1)) is (O(2), (Sym 2 R * )(1), K 1 )where K 1 is described as follows:0 Sym 3 R * R ⊗ Sym 2 R * (1) = R * ⊗ Sym 2 R * R * (1) 0 Sym 3 R * V ⊗ Sym 2 R * (1) K 1 0 (R ⊥ ) * ⊗ Sym 2 R * (1) ( R ⊥ ) * ⊗ Sym 2 R * (1)…”

Involution surface bundles over surfaces

“…Let K be a field of characteristic different from 2. An involution surface X over K is classified by the data of anétale K-algebra L of degree 2 and a central simple algebra B over L of degree 2; see, e.g., [1,Exa. 3.3].…”

“…In this paper, we continue our investigation of del Pezzo fibrations, initiated in [7]. Here we treat the case of fibrations in minimal del Pezzo surfaces of degree 8, also known as involution surfaces and described from the arithmetic viewpoint, e.g., in [1]. These include, as a special case, fibrations in quadric surfaces.…”