Del Pezzo surfaces with infinite automorphism groups

Abstract

We classify del Pezzo surfaces with Du Val singularities that have infinite automorphism groups and describe the connected components of their automorphisms groups. Everywhere below, the number n 0 refers to the corresponding surface in Table 3 in Section 8. As a consequence of our classification we have the following.

Compactifications of the affine plane over non-closed fields

“…If n = 3, then we see ν ′ = ν and ♯ ∆ ′ = 2. Moreover, Γ ′ 0 is a 0-curve by Lemma 2.5 (3). Therefore, we see that the weighted dual graph of ∆ k is given as (28) or (29) in Appendix A.3.…”

“…Then we see that ( Γ ′ 0,1 • ∆ ′ − Γ ′ 0,1 ) = 1 and ∆ ′ is not a linear chain so that Γ ′ 0,1 is a (−1)-curve. Moreover, r > 2 by Lemma 3.8 (3). Namely, the weighted dual graph of ∆ ′ is as follows:…”

“…In §5, we will prove Theorem 1.2 (3). In §6, we will give various remarks about Theorem 1.2 (3). Moreover, we will yield a criterion for del Pezzo fibrations with canonical singularities to contain vertical A 2 -cylinders in terms of degree and singularities of generic fibers (see Theorem 6.7) as an application of Theorem 1.2(3).…”

“…In §4, we will prove Theorem 1.2(1) and ( 2), which are the key results for classifying lc compactifications of the affine plane defined over an algebraically non-closed field. In §5, we will prove Theorem 1.2 (3). In §6, we will give various remarks about Theorem 1.2 (3).…”

Cylinders in Fano varieties

“…Let S be a smooth cubic surface in P 3 . Then any automorphism of the affine variety P 3 n S is induced by an automorphism of P 3 , i.e., we have Aut P 3 n S D Aut P 3 ; S : that admits an effective G a -action (see [41,147,197]). Then Aut.P 3 ; S / contains a subgroup isomorphic to G a , so that Aut.P 3 n S / also contains a subgroup isomorphic to G a .…”