The Largest Automorphism Group of a Del Pezzo Surface of Degree $$2$$ without Points

We apply Nadel’s method of multiplier ideal sheaves to show that every complex del Pezzo surface of degree at most six whose automorphism group acts without fixed points has a Kahler–Einstein metric. In particular, all del Pezzo surfaces of degree 4, 5, or 6 and certain special del Pezzo surfaces of lower degree are shown to have a Kahler–Einstein metric. These existence statements are not new, but the proofs given in the present paper are less involved than earlier ones by Siu, Tian and Tian–Yau.

In this note, which has little pretence to originality, we clarify the relation between the geometry of del Pezzo surfaces of degree 4 and their realization as the zero set of two quadratic forms in five variables. We also review the classical description of the desingularized Kummer surface K constructed from the Jacobian J of a curve C of genus 2 as the zero set of three quadratic forms in six variables (Plucker, Kummer, Klein [7], [6], see [5] or [3] for a modern treatment). If C has a rational Weierstrass point, a partial diagonalization of this system gives rise to a natural projection onto a hyperplane, defining a finite morphism π : K → X of degree 2 onto a del Pezzo surface X of degree 4 (see [9, §6]). We show that X is the blow-up of Pk in the images of the five other Weierstrass points of C under the embedding of Pk as a conic in Pk. The morphism π sends the 16 lines on K to the 16 lines on X, and is equivariant with respect to the action of the subgroup of 2-division points J [2] ⊂ J . Thus π gives rise to a morphism from the twisted Kummer surface to the twisted del Pezzo surface. In our presentation it is obvious that all del Pezzo surfaces of degree 4 can be obtained in this way, an observation made by Victor Flynn in [4]. The fact that any 2-covering of J maps to a del Pezzo surface of degree 4 was first observed in [2], and used in [2], [1] and [9] to construct and visualize elements of order 2 in the Tate–Shafarevich group of J over Q using the theory of the Brauer– Manin obstruction on del Pezzo surfaces of degree 4. It was the author’s desire to understand the geometry behind these calculations that prompted him to write this note. I would like to thank Igor Dolgachev for useful discussions.

In this paper we study quotients of del Pezzo surfaces of degree four and more over arbitrary field k \Bbbk of characteristic zero by finite groups of automorphisms. We show that if a del Pezzo surface X X contains a point defined over the ground field and the degree of X X is at least five, then the quotient is always k \Bbbk -rational. If the degree of X X is equal to four, then the quotient can be non- k \Bbbk -rational only if the order of the group is 1 1 , 2 2 , or 4 4 . For these groups we construct examples of non- k \Bbbk -rational quotients.

This paper explores the computation of the Brauer-Manin obstruction on Del Pezzo surfaces of degree 2, with examples coming from the class of “semi-diagonal” Del Pezzo surfaces of degree 2. It is conjectured that the failure of the Hasse principle for a broad class of varieties, including Del Pezzo surfaces, can always be explained by a nontrivial Brauer-Manin obstruction. We provide computational evidence in support of this conjecture for semi-diagonal Del Pezzo surfaces of degree 2. In addition, we determine the complete list of the possibilities for the finite abelian group H(k,PicX), where X is a Del Pezzo surface of any degree, thus completing a computation which had been previously carried out in various special cases only.

We prove, via an “arithmetic surjectivity” approach inspired by work of Denef, that weak weak approximation holds for surfaces with two conic fibrations satisfying a general assumption. In particular, weak weak approximation holds for general del Pezzo surfaces of degrees $1$ or $2$ with a conic fibration.

— We survey the state of affairs for the distribution of Q-rational points on non-singular del Pezzo surfaces of low degree, highlighting the recent resolution of Manin’s conjecture for a non-singular del Pezzo surface of degree 4 by la Breteche and Browning [3].

Let [Formula: see text] be any field of characteristic zero, [Formula: see text] be a del Pezzo surface and [Formula: see text] be a finite subgroup in [Formula: see text]. In this paper, we study when the quotient surface [Formula: see text] can be non-rational over [Formula: see text]. Obviously, if there are no smooth [Formula: see text]-points on [Formula: see text] then it is not [Formula: see text]-rational. Therefore, under assumption that the set of smooth [Formula: see text]-points on [Formula: see text] is not empty we show that there are few possibilities for non-[Formula: see text]-rational quotients. The quotients of del Pezzo surfaces of degree [Formula: see text] and greater are considered in the author’s previous papers. In this paper, we study the quotients of del Pezzo surfaces of degree [Formula: see text]. We show that they can be non-[Formula: see text]-rational only for the trivial group or cyclic groups of order [Formula: see text], [Formula: see text] and [Formula: see text]. For the trivial group and the group of order [Formula: see text], we show that both [Formula: see text] and [Formula: see text] are not [Formula: see text]-rational if the [Formula: see text]-invariant Picard number of [Formula: see text] is [Formula: see text]. For the groups of order [Formula: see text] and [Formula: see text], we construct examples of both [Formula: see text]-rational and non-[Formula: see text]-rational quotients of both [Formula: see text]-rational and non-[Formula: see text]-rational del Pezzo surfaces of degree [Formula: see text] such that the [Formula: see text]-invariant Picard number of [Formula: see text] is [Formula: see text]. As a result of complete classification of non-[Formula: see text]-rational quotients of del Pezzo surfaces we classify surfaces that are birationally equivalent to quotients of [Formula: see text]-rational surfaces, and obtain some corollaries concerning fields of invariants of [Formula: see text].

The blow-up of the anticanonical base point on a del Pezzo surface S of degree 1 gives rise to a rational elliptic surface $\mathscr {E}$ with only irreducible fibers. The sections of minimal height of $\mathscr {E}$ are in correspondence with the $240$ exceptional curves on S. A natural question arises when studying the configuration of these curves: if a point on S is contained in “many” exceptional curves, is it torsion on its fiber on $\mathscr {E}$ ? In 2005, Kuwata proved for the analogous question on del Pezzo surfaces of degree $2$ , where there are 56 exceptional curves, that if “many” equals $4$ or more, the answer is yes. In this paper, we prove that for del Pezzo surfaces of degree 1, the answer is yes if ‘many’ equals $9$ or more. Moreover, we give counterexamples where a non-torsion point lies in the intersection of $7$ exceptional curves. We give partial results for the still open case of 8 intersecting exceptional curves.

We investigate a smooth del Pezzo surface of degree 2 which is not Frobenius split. We give a characterization of non-F-split del Pezzo surfaces of degree 2 which exist only if the characteristic of the ground field is 2 or 3. Moreover, we prove that the set of centers of the blow-ups on $\mathbb{P}^{2}$ which gives a non-F-split del Pezzo surface is projectively equivalent to the only complete 7-arc over $\mathbb{F}_{9}$ if the characteristic is 3.

Construction of a del Pezzo surface with maximal Galois action on its Picard group

Various ample divisors on smooth del Pezzo surfaces of degree [Formula: see text] have been verified to allow polar cylinders in [I. Cheltsov, J. Park and J. Won, Cylinders in del Pezzo surfaces, Int. Math. Res. Not. 2017(4) (2015) 1179–1230]. We show that affine cones over smooth del Pezzo surfaces of degree [Formula: see text] polarized by such ample divisors are flexible in codimension one. All varieties in this paper are assumed to be defined over an algebraically closed field of characteristic zero.

We obtain necessary and sufficient condition for the existence of del Pezzo surfaces of degrees $5$ and $6$ over a field $K$ with a prescribed action of absolute Galois group $\operatorname{Gal}(K^{\mathrm{sep}}/K)$ on the graph of $(-1)$-curves. We also compute the automorphism groups of del Pezzo surfaces of degree $5$ over arbitrary fields. Bibliography: 19 titles.

Among geometrically rational surfaces, del Pezzo surfaces of degree two over a field k containing at least one point are arguably the simplest that are not known to be unirational over k. Looking for k-rational curves on these surfaces, we extend some earlier work of Manin on this subject. We then focus on the case where k is a finite field, where we show that all except possibly three explicit del Pezzo surfaces of degree two are unirational over k.

Let $\Bbbk$ be any field of characteristic zero, $X$ be a del Pezzo surface of degree~$2$ and $G$ be a group acting on $X$. In this paper we study $\Bbbk$-rationality questions for the quotient surface $X / G$. If there are no smooth $\Bbbk$-points on $X / G$ then $X / G$ is obviously non-$\Bbbk$-rational. Assume that the set of smooth $\Bbbk$-points on the quotient is not empty. We find a list of groups, such that the quotient surface can be non-$\Bbbk$-rational. For these groups we construct examples of both $\Bbbk$-rational and non-$\Bbbk$-rational quotients of both $\Bbbk$-rational and non-$\Bbbk$-rational del Pezzo surfaces of degree $2$ such that the $G$-invariant Picard number of $X$ is $1$. For all other groups we show that the quotient $X / G$ is always $\Bbbk$-rational.

Let $X$ be a del Pezzo surface of degree $5$ defined over a field $F$. A theorem of Yu. I. Manin and P. Swinnerton-Dyer asserts that every Del Pezzo surface of degree $5$ is rational. In this paper we generalize this result as follows. Recall that del Pezzo surfaces of degree $5$ over a field $F$ are precisely the twisted $F$-forms of the moduli space $\overline{M_{0, 5}}$ of stable curves of genus $0$ with $5$ marked points. Suppose $n \geq 5$ is an integer, and $F$ is an infinite field of characteristic $\neq 2$. It is easy to see that every twisted $F$-form of $\overline{M_{0, n}}$ is unirational over $F$. We show that (a) if $n$ is odd, then every twisted $F$-form of $\overline{M_{0, n}}$ is rational over $F$. (b) If $n$ is even, there exists a field extension $F/k$ and a twisted $F$-form $X$ of $\overline{M_{0, n}}$ such that $X$ is not retract rational over $F$.