aa r X i v : . [ m a t h . AG ] M a r DEL PEZZO SURFACES OVER FINITE FIELDS
ANDREY TREPALIN
Abstract.
Let X be a del Pezzo surface of degree 2 or greater over a finite field F q . Theimage Γ of the Galois group Gal( F q / F q ) in the group Aut(Pic( X )) is a cyclic subgrouppreserving the anticanonical class and the intersection form. The conjugacy class of Γin the subgroup of Aut(Pic( X )) preserving the anticanonical class and the intersectionform is a natural invariant of X . We say that the conjugacy class of Γ in Aut(Pic( X )) isthe type of a del Pezzo surface. In this paper we study which types of del Pezzo surfacesof degree 2 or greater can be realized for given q . We collect known results about thisproblem and fill the gaps. Introduction A del Pezzo surface is a smooth projective surface such that its anticanonical classis ample. Let X be a del Pezzo surface over finite field F q , and X = X ⊗ F q . TheGalois group Gal (cid:0) F q / F q (cid:1) acts on the lattice Pic( X ) and preserves the anticanonicalclass − K X and the intersection form. The image Γ of Gal (cid:0) F q / F q (cid:1) is a cyclic sub-group in Aut (cid:0) Pic( X ) (cid:1) preserving the intersection form. Obviously, the set of elements inAut (cid:0) Pic( X ) (cid:1) preserving the intersection form is a group. We denote this group by W .The group Γ is a cyclic subgroup of W . Therefore the conjugacy class of Γ in W is anatural invariant of a del Pezzo surface. We say that the conjugacy class of Γ in W is the type of a del Pezzo surface. It is well-known (see, for example, [Man74, Theorem IV.1.1])that the type of a del Pezzo surface defines the number N of F q -points on X . One has N = q + aq + 1 , (1.1)where a is the trace of a generator g of Γ considered as an element on GL(Pic( X )).More generally, let N k be the number of F q k points of X k = X ⊗ F q k . The zeta functionof X is the formal power series Z X ( t ) = exp ∞ X k =1 N k t k k ! . For a geometrically rational surface X one has (see [Man74, Corollary 2 from Theo-rem IV.5.1]) Z X ( t ) = 1(1 − t )(1 − q t ) det(1 − qtg | Pic( X ) ⊗ Q ) . Therefore the zeta function of a del Pezzo surface X is totally defined by the type of X .Thus the type of X defines the numbers N k , since these numbers are uniquely determinedby a given zeta function. The research was carried out at the IITP RAS at the expense of the Russian Foundation for Sciences(project N ◦ he study of algebraic varieties over finite fields has numerous applications in codingtheory (see, for example, [VNT18]).The natural question is which types of del Pezzo surfaces can be realized for given q .In [BFL16, Theorem 1.7] it is shown that a del Pezzo surface of any type exists for anysufficiently big q . But there is no answer to the question for small q . The aim of thispaper is to construct each type of del Pezzo surfaces of degree 2 or greater over F q if it ispossible, and show that such surfaces do not exist for other values of q .The degree of a del Pezzo surface X is the number d = K X . One has 1 d
9. Foreach d we give the isomorphism class of W in the following table, where W ( E ), W ( E )and W ( E ) are the Weyl groups of the corresponding root systems E , E and E .Degree 9 8 7 6 5 4 3 2 1Group h id i h id i or Z / Z Z / Z D S ( Z / Z ) ⋊ S W ( E ) W ( E ) W ( E )One can easily find conjugacy classes of elements in the dihedral group D of order 12and the symmertic group S . For the group ( Z / Z ) ⋊ S it is more difficult, and we give aproof of the classification in Proposition 6.1, since in some sources there are missed casesin the classification of finite subgroups of ( Z / Z ) ⋊ S . The classification of conjugacyclasses of the elements in the groups W ( E ), W ( E ) and W ( E ) is obtained in [Fr51]and [Fr67].A surface S is called minimal if any birational morphism S → S ′ is an isomorphism. Thetype of a del Pezzo surface X allows to determine whether the surface is minimal or not.If X is not a minimal surface then it is isomorphic to the blowup of a surface Y at somepoints of certain degrees. Such points of the blowup should satisfy some conditions thatare called being in general position (see Theorem 2.2, Proposition 6.4 and Corollary 2.6).An important case of nonminimal del Pezzo surfaces is the blowup of the projective planeat some points of certain degrees in general position. Surfaces of these types have manyinteresting properties and are studied in many papers. For example, the blowups of P F q at six F q -points or four F q -points and a point of degree two are considered in [SD10],and the blowup of P F q at seven F q -points is considered in [Kap13, Chapter 4, Sections 2and 3] and [KR16] for odd q . Many types (including most complicated) of nonminimaldel Pezzo surfaces are considered in [BFL16], but the aim of [BFL16] is to construct delPezzo surfaces with a given number of F q -points. Some types of nonminimal del Pezzosurfaces are not considered in [BFL16] because del Pezzo surfaces of different types canhave the same number of F q -points.The types of minimal del Pezzo surfaces of degrees 4, 3 and 2 are considered in thepapers of S. Rybakov and the author. Minimal del Pezzo surfaces of degree 4 are con-structed in [Ry05, Theorem 3.2]. These surfaces either admit a structure of a conic bundlewith four degenerate fibres, or are isomorphic to the contracting of a ( − P F q . There are five types ofminimal cubic surfaces. One type is constructed in [SD10] and the other four types areconstructed in [RT17, Theorem 1.2], but there are some restrictions on q for three types.Minimal del Pezzo surfaces of degree 2 are constructed in [Tr17]. For four types there aresome restrictions on q (three of them are inherited from the restrictions on q for the caseof cubic surfaces).We give the classification of the types of del Pezzo surfaces of degree 5 or greater, 4, 3and 2 in Tables 1, 4, 3 and 2 respectively. he main results of this paper are the following three theorems. Theorem 1.2.
In the notation of Table 2 the following holds. (1)
Del Pezzo surfaces of degree of types and do not exist over F , F , F , F , F , F , and exist over other finite fields. (2) Del Pezzo surfaces of degree of types and do not exist over F , F , F , F ,and exist over other finite fields. (3) Del Pezzo surfaces of degree of types , , and do not exist over F , F , F , and exist over other finite fields. (4) Del Pezzo surfaces of degree of types , – , – , , , , , , , , , , , , do not exist over F , and exist over other finite fields. (5) Del Pezzo surfaces of degree of types , , , , , , , , , , , , , , , – , , , , , – exist over all finite fields. (6) Del Pezzo surfaces of degree of types and do not exist over F , and existover any F q where q > . (7) Del Pezzo surfaces of degree of types and exist over any F q where q = 6 k +1 . Theorem 1.3.
In the notation of Table 3 the following holds. (1)
A cubic surface of type ( c ) does not exist over F , F , F , and exists over otherfinite fields. (2) Cubic surfaces of types ( c ) , ( c ) do not exist over F , F , and exist over otherfinite fields. (3) Cubic surfaces of types ( c ) , ( c ) , ( c ) , ( c ) do not exist over F , and exist overother finite fields. (4) Cubic surfaces of types ( c ) – ( c ) , ( c ) – ( c ) and ( c ) – ( c ) exist over all finitefields. (5) A cubic surface of type ( c ) exists over any F q where q = 6 k + 1 . Theorem 1.4.
In the notation of Table 4 the following holds. (1)
Del Pezzo surfaces of degree of types id , ι ab , ι abcd do not exist over F , F , andexist over other finite fields. (2) Del Pezzo surfaces of degree of types ( ab )( cd ) , ( ab )( cd ) ι ac , ( ab )( cd ) ι ae do notexist over F , and exist over other finite fields. (3) The other types of del Pezzo surfaces of degree exist over all finite fields. The plan of this paper is as follows.In Section 2 we remind some notions about del Pezzo surfaces and the correspondingWeyl groups.In Section 3 we give a short proof of a well-known fact, that del Pezzo surfaces ofdegree 5 or greater of any type exist over any F q , and give the classification of the typesof del Pezzo surfaces of degree 5 or greater.In Sections 4–6 we prove Theorems 1.2, 1.3 and 1.4. By Corollary 2.6 if a del Pezzosurface Y of degree d > F q -point not lying on the lines, then the blowup of Y at this point a del Pezzo surface X of degree d −
1. Moreover, the type of X is definedby the type of Y . Therefore surfaces of the same type as Y exist over any finite field, suchthat there exists a surface X of the corresponding type. Thus our strategy is the following.We start from del Pezzo surfaces of degree 2, then we pass to del Pezzo surfaces of degree 3and consider only those types, for which del Pezzo surfaces of the corresponding types do ot exist over all finite fields. After this we consider the types of del Pezzo surfaces ofdegree 4 such that their existence over all finite fields does not immediately follows fromthe existence of del Pezzo surface of degree 3 of the corresponding types.In Section 4 we consider del Pezzo surfaces of degree 2 and prove Theorem 1.2. InTable 2 we give the classification of conjugacy classes in W ( E ). These 60 conjugacyclasses are divided into 30 pairs, such that the Geiser twist (see Definition 4.2) of asurface of one type in a pair has the other type in the pair. The surfaces of types, that arepaired by Geiser twist, exist over the same finite fields. Many pairs of types are consideredin [BFL16] and [Tr17]. For each of the remaining pairs of types one type in the pair can beconstructed as either the blowup of P F q at some points of certain degrees, or the blowup ofa minimal del Pezzo surface of degree 4 at two F q -points, or the blowup of a cubic surfaceat an F q -point. We give the constructions of del Pezzo surfaces of degree 2 of these typesover F q if it is possible, and show that such types of surfaces do not exist for other valuesof q .In Section 5 we consider del Pezzo surfaces of degree 3 (that are cubic surfaces) andprove Theorem 1.3. In Table 3 we give the classification of conjugacy classes in W ( E ).There are 25 types of cubic surfaces. Existence of cubic surfaces of eight types overall finite fields immediately follows from the existence of the corresponding (see the lastcolumn of Table 3) del Pezzo surfaces of degree 2 over all finite fields. Many types areconsidered in [Ry05], [SD10], [RT17], [Tr17] and [BFL16]. Some types of cubic surfacesare considered in the proofs of some lemmas in Section 4. The remaining three typescan be constructed as the blowup of P F q at some points of certain degrees. We give theconstructions of cubic surfaces of these types over F q if it is possible, and show that suchtypes of surfaces do not exist for other values of q .In Section 6 we consider del Pezzo surfaces of degree 4 and prove Theorem 1.4. InTable 4 we give the classification of conjugacy classes in W ( D ) ∼ = ( Z / Z ) ⋊ S . Thereare 18 types of del Pezzo surfaces of degree 4. For almost all types del Pezzo surfacesof degree 4 exist over all finite fields, since cubic surfaces of the corresponding (see thelast column of Table 4) types exist over all finite fields. Three of the remaining types areconsidered in [Ry05] and [BFL16]. The remaining three types can be constructed as theblowup of P F q or a smooth quadric in P F q at some points of certain degrees. We give theconstructions of del Pezzo surfaces of degree 4 of these types over F q if it is possible, andshow that such types of surfaces do not exist for other values of q .In Section 7 we discuss problems arising for del Pezzo surfaces of degree 1, and showthat for almost all methods of constructing of del Pezzo surfaces of degree 2 or greaterthere appear some additional difficulties.The author is a Young Russian Mathematics award winner and would like to thank itssponsors and jury. Also the author is grateful to S. Rybakov for valuable comments. Notation.
Throughout this paper all surfaces are smooth, projective and defined overa finite field F q of order q , where q is a power of prime. For a surface X we de-note X ⊗ F q by X . For a del Pezzo surface X we denote the Picard group by Pic( X ),the image of the group Gal (cid:0) F q / F q (cid:1) in the corresponding Weyl group W ( R ) acting onthe Picard group Pic( X ) is denoted by Γ, for the Γ-invariant Picard group Pic( X ) Γ onehas Pic( X ) Γ = Pic( X ). The number ρ ( X ) = rk Pic( X ) is the Picard number of X , forthe Γ-invariant Picard number ρ ( X ) Γ one has ρ ( X ) Γ = ρ ( X ). The subspace of classes C in Pic( X ) ⊗ Q such that C · K X = 0 is denoted by K ⊥ X . or any variety X we denote by F the Frobenius automorphism of X .We denote by ξ d a primitive root of unity of order d , and ω = ξ , i = ξ .2. Del Pezzo surfaces and Weyl groups
In this section we remind some basic notions about del Pezzo surfaces and the corre-sponding Weyl groups.
Definition 2.1.
A smooth projective surface X such that the anticanonical class − K X is ample is called a del Pezzo surface .The number d = K X is called the degree of a del Pezzo surface.It is well known that a del Pezzo surface X over an algebraically closed field k isisomorphic to P k , P k × P k or the blowup of P k at up to 8 points in general position . Moreprecisely, the following theorem holds. Theorem 2.2 (cf. [Man74, Theorem IV.2.5]) . Let d , and p , . . . , p − d be − d points on the projective plane P k such that • no three lie on a line; • no six lie on a conic; • for d = 1 the points are not on a singular cubic curve with singularity at one ofthese points.Then the blowup of P k at p , . . . , p − d is a del Pezzo surface of degree d .Moreover, any del Pezzo surface X of degree d over algebraically closed field k is the blowup of such set of points. Definition 2.3.
We say that a collection of geometric points on P k is in general position if it satisfies the conditions of Theorem 2.2.Moreover, for a number of points of certain degree on P F q we say that these points are in general position if the corresponding geometric points are in general position. Definition 2.4.
A curve E of genus 0 on a surface such that E = − − -curve .For a surface X one has ( K X + E ) · E = −
2, thus if X is a del Pezzo surfacethen − K X · E = 1, and the anticanonical morphism ϕ |− K X | maps E to a line in a projec-tive space of dimension d = K X for d > − lines .The following theorem describes the set of ( − Theorem 2.5 (cf. [Man74, Theorem IV.4.3]) . Let X be a del Pezzo surface of degree d and f : X → P k be the blowup of P k at points p , . . . , p − d . Then the set of ( − -curves on X consists of the following curves: • the preimages of p i ; • the proper transforms of lines passing through two points from the set { p i } ; • the proper transforms of conics passing through five points from the set { p i } ; • the proper transforms of cubics passing through seven points from the set { p i } andpassing through one of these points with multiplicity ; • the proper transforms of quartics passing through eight points from the set { p i } and passing through three of these points with multiplicity ; the proper transforms of quintics passing through eight points from the set { p i } and passing through six of these points with multiplicity ; • the proper transforms of sextics passing through eight points from the set { p i } withmultiplicity at least and passing through one of these points with multiplicity . Applying Theorem 2.5 one can compute the numbers of ( − Corollary 2.6.
Let X be a del Pezzo surface of degree d , and let p be an F q -pointthat does not lie on the lines. Then the blowup of X at p is a del Pezzo surface ofdegree d − . To apply Corollary 2.6 one should compare the number of F q -points on X and thenumber of F q -points on X lying on the lines. The number of F q -points on X of certaintype is given by equation (1.1). The following remark allows us to compute the numberof F q -points on X lying on the lines. Remark . Let X be a del Pezzo surface of degree 3 d
7. Then any line definedover F q contains q + 1 points defined over F q .If d > F q -points on X lying on the lines is equal to A ( q + 1) − B + C , where A isthe number of lines defined over F q , B is the number of pairs of meeting each other linesdefined over F q , and C is the number of pairs of meeting each other conjugate lines definedover F q .If d = 3 then then there are no points of intersection of four or more lines. Threelines H , H and H meet each other, if and only if the divisor H + H + H is linearlyequivalent to − K X . For such triple of lines there are two possibilities: either there arethree distinct points H ∩ H , H ∩ H , H ∩ H , or the three lines have a common point,that is called an Eckardt point . Therefore if all three lines are defined over F q , then theunion of these lines contains 3 q or 3 q + 1 points defined over F q . If one of these linesis defined over F q , and the others are conjugate and defined over F q , then the union ofthese lines contains q + 2 or q + 1 points defined over F q . If these three lines are conjugateand defined over F q , then the union of these lines contains 0 or 1 point defined over F q .In all other cases there are no F q -points on the union of these three lines.Therefore for d = 3 the number of F q -points on X lying on the lines is equalto A ( q + 1) − B + C + D − E + F , where A is the number of lines defined over F q , B is the number of pairs of meeting each other lines defined over F q , C is the number ofpairs of meeting each other conjugate lines defined over F q , D is the number of Eckardtpoints lying on a triple of lines defined over F q , E is the number of Eckardt points lyingon a triple of lines defined over F q such that two of these lines are conjugate, F is thenumber of Eckardt points lying on a triple of conjugate lines defined over F q . Note thatthe type of a cubic surface X defines the numbers A , B and C . Therefore the num-ber of F q -points on X lying on the lines is at least A ( q + 1) − B + C − E and at most A ( q + 1) − B + C + D + F . o count the mentioned in Remark 2.7 numbers of Eckardt points we need the followinglemma. Lemma 2.8 ([DD17, Lemma 9.4]) . A line on a cubic surface contains , or Eckardtpoints if q is odd, and contains , or Eckardt points if q is even. If X is not isomorphic to P k × P k , then the Picard group of X is generated by the propertransform L of the class of a line on P k , and the classes E , . . . , E − d of the exceptionaldivisors. For d K ⊥ X ⊂ Pic (cid:0) X (cid:1) ⊗ Q is generated by L − E − E − E , E − E , E − E , . . . , E − d − E − d . This set of generators are simple roots for the root system of a certain type given in thefollowing table (for details see e. g. [Man74, Theorem IV.3.5]).Degree 6 5 4 3 2 1Root system A × A A D E E E To simplify the notation we denote by E − d the root system corresponding to a delPezzo surface of degree d .Any group, acting on the Picard lattice Pic (cid:0) X (cid:1) and preserving the intersection form,is a subgroup of the Weyl group W ( E − d ). In particular, if X is a del Pezzo surface overa field k , then the group Gal (cid:0) k / k (cid:1) acts on Pic( X ), and its image Γ in Aut (cid:0) Pic (cid:0) X (cid:1)(cid:1) is a subgroup of W ( E − d ). Moreover, if k is a finite field then Γ is a cyclic subgroupof W ( E − d ). One can easily classify finite subgroups of W ( A × A ) ∼ = D that is adihedral group of order 12, W ( A ) ∼ = S and W ( D ) ∼ = ( Z / Z ) ⋊ S . The classificationof finite subgroups in W ( E ), W ( E ), W ( E ) is obtained in [Car72].One of the main properties of a conjugacy class in W ( E − d ) is a Carter graph,that was introduced in [Car72]. This graph describes eigenvalues of the action of Γon K ⊥ X ⊂ Pic (cid:0) X (cid:1) ⊗ Q (see [Car72, Table 3]), and, in particular, defines the order of Γ.Moreover, we have the following useful well-known lemma. Lemma 2.9.
Let Y be a del Pezzo surface of type whose Carter graph is R , and X bethe blowup of Y at a point P of degree k . If X is a del Pezzo surface then its type hasthe Carter graph R × A k − . In particular, if P is an F q -point then the types of X and Y have the same Carter graph. Del Pezzo surfaces of degree and greater In this section we give a proof of the well-known fact that del Pezzo surfaces of degree 5or greater of any types exist over all finite fields. For the sake of completeness we givethe classification of the types of del Pezzo surface of degree 5 or greater.
Proposition 3.1.
Del Pezzo surfaces of degree or greater of any types exist over allfinite fields.Proof. For del Pezzo surfaces of degree 5 the group Γ is a cyclic subgroup of the Weylgroup W ( A ) ∼ = S . The conjugacy class of a cyclic subgroup Γ in S is defined by thecyclic type of σ ∈ S .For a given σ consider a smooth conic Q on P F q and five geometric points p , . . . , p on Q ,such that F acts on these points as p i p σ ( i ) . Consider the blowup f : X → P F q of thepoints p , . . . , p . The surface X is a del Pezzo surface of degree 4 by Theorem 2.2, since urface Class Graph Order Eigenvalues ρ ( X ) Γ P id ∅ X id ∅ P × P id ∅ P × P g A − X id ∅ X g A − X id ∅ X s A − X rs A − − X r A − X r A ω , ω X r A × A ω , ω , − X id ∅ X ( ab ) A − X ( ab )( cd ) A − − X ( abc ) A ω , ω X ( abcd ) A − − i 2 X ( abcde ) A ξ , ξ , ξ , ξ X ( abc )( de ) A × A ω , ω , − Table 1.
Types of del Pezzo surfaces of degree 5 or greaterif three points p i lie on a line l then l · Q > f − ∗ ( Q ) = − π : X → X of f − ∗ ( Q ), where X is a del Pezzosurface of degree 5 of type corresponding to σ .This method works in all cases except the trivial group and the group Z / Z ∼ = h (12)(34) i over F , since there are only three F -points and one point of degree 2 on a conic. Butin these cases one can consider a blowup X → P F at four F -points or two points ofdegree 2 in general position respectively.On a del Pezzo surface of degree 6–9 one can blow up a number of F q -points and geta del Pezzo surface of degree 5. One can consider a del Pezzo surface of degree 5 of thecorresponding type and contract a number of ( − F q . (cid:3) In Table 1 for del Pezzo surfaces X of degree 5 and greater we give the classification ofconjugacy classes in the groups acting on Pic( X ) and preserving K X and the intersectionform. The first column is the isomorphism class of the surface X = X ⊗ F q . In thiscolumn we denote by X d del Pezzo surfaces such that X d can be obtained as the blowupof P F q at 9 − d points. The second column is a conjugacy class of the generator of Γ. Wedenote the generator of Z / Z by g , and denote the generators of D by r and s , with therelations r = s = srsr = 1. The third column is a Carter graph corresponding to theconjugacy class (see [Car72]). The fourth column is the order of an element. The fifthcolumn is the collection of eigenvalues of the action of an element on K ⊥ X ⊂ Pic( X ) ⊗ Q .The sixth column is the invariant Picard number ρ ( X ) Γ . . Del Pezzo surfaces of degree ⊂ W ( E ) we construct the corresponding delPezzo surface of degree 2 over F q if it is possible, and show that such surfaces do not existfor other values of q . These results are summed up in Theorem 1.2.In Table 2 we collect some facts about conjugacy classes of elements in the Weylgroup W ( E ). This table is based on [Car72, Table 10]. The first column is the number ofa conjugacy class in order of their appearence in Carter’s table. The second column is aCarter graph corresponding to the conjugacy class (see [Car72]). The third column is theorder of an element. The fourth column is the collection of eigenvalues of the action of anelement on K ⊥ X ⊂ Pic( X ) ⊗ Q . The fifth column is the invariant Picard number ρ ( X ) Γ .The last column is the number of the corresponding conjugacy class after the Geiser twist(see Definition 4.2).Number Graph Order Eigenvalues ρ ( X ) Γ Geiser1. ∅ A − A − − A ω , ω A − − − A − − − A × A − ω , ω A − − i 5 33.9. A − − − − A − − − − A × A − − ω , ω A ω , ω , ω , ω A × A − − i, − A × A − − i, − A ξ , ξ , ξ , ξ D − − ω , − − ω D ( a ) 4 1, 1, 1, i, − i, i, − i 4 50.18. A − − − − − A × A − − − ω , ω A × A − ω , ω , ω , ω A × A − − i, − − A × A − − i, − − A × A
12 1, 1, i, − − i, ω , ω A × A
10 1, 1, ξ , ξ , ξ , ξ , − A − ω , ω , − ω , − ω A − ω , ω , − ω , − ω D × A − − ω , − − ω , − D ( a ) × A − i, i, − i, − D − ξ , ξ , ξ , ξ D ( a ) 12 1, 1, i, − i, − ω , − − ω A − − − − − − umber Graph Order Eigenvalues ρ ( X ) Γ Geiser32. A ω , ω , ω , ω , ω , ω A × A − − i, − − − A × A × A
12 1, i, − − i, ω , ω , − A − − i, i, − − i 2 28.36. A × A
15 1, ξ , ξ , ξ , ξ , ω , ω A × A − ω , ω , − ω , − ω , − A × A − ω , ω , − ω , − ω , − A ξ , ξ , ξ , ξ , ξ , ξ D × A − − ω , − − ω , − − D × A − ξ , ξ , ξ , ξ , − D ( a ) × A
12 1, i, − i, − ω , − − ω , − D
10 1, − − ξ , − ξ , − − ξ , − ξ D ( a ) 8 1, i, − i, ξ , ξ , ξ , ξ D ( a ) 6 1, − ω , − − ω , − ω , − − ω E
12 1, ω , ω , − i ω , − i ω , i ω , i ω E ( a ) 9 1, ξ , ξ , ξ , ξ , ξ , ξ E ( a ) 6 1, ω , ω , − ω , − ω , − ω , − ω A − − − − − − − A × A −
1, i, − − i, i, − − i 1 17.51. A × A ω , ω , − ω , ω , − ω , − ω A ξ , i, ξ , − ξ , − i, ξ D × A − − − − − ω , − − ω D × A − − − − ξ , − ξ , − ξ , − ξ D ( a ) × A − − ω , − − ω , − ω , − − ω E − − ξ , − ξ , − ξ , − ξ , − ξ , − ξ E ( a ) 14 − ξ , − ξ , − ξ , − − ξ , − ξ , − ξ E ( a ) 12 − ω , − − ω , − i ω , − i ω , i ω , i ω E ( a ) 30 − ω , − − ω , − ξ , − ξ , − ξ , − ξ E ( a ) 6 − ω , − − ω , − ω , − ω , − ω , − ω W ( E )The Geiser twist of a del Pezzo surface of degree 2 is very useful in what follows. Itwas used in [Tr17, Section 3] and [BFL16, Subsection 4.1.2.]. We recall some notionsintroduced in [Tr17, Section 3].For a del Pezzo surface X of degree 2 the anticanonical linear system | − K X | gives adouble cover X → P F q . This cover defines an involution on X that is called the Geiserinvolution . Therefore we can apply the following well-known proposition (see, for example,[RT17, Proposition 4.4]). Proposition 4.1.
Let X be a smooth algebraic variety over a finite field F q such thata cyclic group G of order n acts on X and this action induces a faithful action of G on the group Pic( X ) . Let Γ be the image of the Galois group Gal (cid:0) F q / F q (cid:1) in thegroup Aut (cid:0)
Pic( X ) (cid:1) . Let h and g be the generators of Γ and G respectively. hen there exists a variety X such that the image Γ of the Galois group Gal (cid:0) F q / F q (cid:1) in the group Aut (cid:0)
Pic( X ) (cid:1) ∼ = Aut (cid:0) Pic( X ) (cid:1) is generated by the element gh . Note that in Proposition 4.1 one has X ∼ = X . Therefore if X is a del Pezzo surface,then X is a del Pezzo surface of the same degree. Definition 4.2 ([Tr17, Definition 3.2]) . Let X be a del Pezzo surface of degree 2 suchthat the image Γ of the Galois group Gal (cid:0) F q / F q (cid:1) in the group Aut (cid:0) Pic( X ) (cid:1) is generatedby an element h . Then by Proposition 4.1 there exists a del Pezzo surface X of degree 2such that the image Γ of the Galois group Gal (cid:0) F q / F q (cid:1) in the group Aut (cid:0) Pic( X ) (cid:1) isgenerated by the element γh . We say that the surface X is the Geiser twist of thesurface X .Therefore 60 types of del Pezzo surfaces of degree 2 are divided into 30 pairs, and asurface of a given type exists if and only if the Geiser twist of this surface exists. Remark . Note that the Geiser involution γ acts on K ⊥ X bymultiplying all elements by −
1. Therefore the eigenvalues of a generator of the group Γ are the eigenvalues of a generator of the group Γ multiplied by −
1. Thus for each typeof the group Γ it is easy to find the type of the corresponding group Γ (see Table 2),except the cases where two types of Γ have the same collections of eigenvalues.We use the following lemma to distinguish types of del Pezzo surfaces with the samecollection of eigenvalues. Lemma 4.4.
If there is a Γ -invariant ( − -curve on a del Pezzo surface X of degree then there are no Γ -invariant ( − -curves on the Geiser twist X of X . In particular, ina pair of del Pezzo surfaces X and X of degree such that X is the Geiser twist of X ,no more than one surface is isomorphic to the blowup of a cubic surface at an F q -point.Proof. If there is a Γ -invariant ( − E on a del Pezzo surface X of degree 2 thenone can contract this curve and get a cubic surface Y . The set of ( − X consistof E , the proper transforms of the 27 lines on Y , and images of these 28 curves underthe action of the Geiser involution. Therefore there are no Γ -invariant pairs of ( − and the Geiser involution on X . Thus there are no Γ -invariant( − X . (cid:3) Corollary 4.5.
There are six pairs of types of del Pezzo surfaces of degree that havethe same Carter graph: and , and , and , and , and , and (see Table 2). For these types we have the following description. • A del Pezzo surface X of type is isomorphic to the blowup of P F q × P F q at threepoints of degree . The Geiser twist X of X has type and is isomorphic tothe blowup of a del Pezzo surface of degree of type ι abcd (see Table 4) at two F q -points. • A del Pezzo surface X of type is isomorphic to the blowup of P F q at an F q -pointand three points of degree . The Geiser twist X of X has type and is isomor-phic to the blowup of a del Pezzo surface Y of degree with ρ ( Y ) = 1 at threepoints of degree . • A del Pezzo surface X of type is isomorphic to the blowup of P F q × P F q at apoint of degree and a point of degree . The Geiser twist X of X has type nd is isomorphic to the blowup of a del Pezzo surface of degree of type ( ab ) ι acde (see Table 4) at two F q -points. • A del Pezzo surface X of type is isomorphic to the blowup of P F q at an F q -point,a point of degree and a point of degree . The Geiser twist X of X has type and is isomorphic to the blowup of a del Pezzo surface Y of degree with ρ ( Y ) = 1 at a point of degree and a point of degree . • A del Pezzo surface X of type is isomorphic to the blowup of P F q × P F q at apoint of degree . The Geiser twist X of X has type and is isomorphic to theblowup of a cubic surface of type c (see Table 3) at an F q -point. • A del Pezzo surface X of type is isomorphic to the blowup of P F q at an F q -pointand a point of degree . The Geiser twist X of X has type and is isomorphicto the blowup of a del Pezzo surface Y of degree with ρ ( Y ) = 1 at a point ofdegree .Remark . In Table 2 there are no data describing difference between types of del Pezzosurfaces of degree 2 with the same Carter graph. One can find these data in [Car72,Table 10] or [Ur96, Table 1] and see that the given in Corollary 4.5 descriptions satisfythese data.
Proof of Corollary 4.5.
The described surfaces of certain types have the correspondingto these types Carter graphs by Lemma 2.9. The surfaces of types 6, 10, 14, 22, 26and 38 are isomorphic to the blowup of certain cubic surfaces at an F q -point. Thereforeby Lemma 4.4 any surface of one of these types cannot be the Geiser twist of an othersurface of one of these types. (cid:3) We prove Theorem 1.2 case by case. We start from the types for which a del Pezzosurface of degree 2 is isomorhic to the blowup of P F q at a number of points of certaindegrees. There are 15 types of such del Pezzo surfaces parametrised by Young diagrams.Seven of these types are considered in [BFL16, Subsection 4.2] and three other types areconsidered in [Tr17]. The remaining five types are considered below.For types 3 and 6 we use the notation of the following remark. Remark . A del Pezzo surface X of degree 2 of type 3 or 6 is isomorphic to the blowupof a cubic surface Y at an F q -point not lying on any line. Moreover, there is a morphism f : Y → P F q that contracts six lines on Y to 6 geometric points p , . . . , p on P F q .By Theorem 2.5 the set of lines on Y consists of E i = f − ( p i ), the proper transforms L ij ∼ L − E i − E j of the lines passing through a pair of points p i and p j , and the propertransforms Q j ∼ L + E j − X i =1 E i of the conics passing through five points p i for i = j . Note that the group S ⊂ W ( E )naturally acts on this set in the following way: for σ ∈ S one has σ ( E i ) = E σ ( i ) , σ ( L ij ) = L σ ( i ) σ ( j ) , and σ ( Q i ) = Q σ ( i ) . For considered types of surfaces the actions of F on the set of lines on Y coincides with the actions of certain permutations. Lemma 4.8.
A del Pezzo surface of degree of type does not exist over F , F , F andexists over other finite fields. roof. If X has type 3 then the Galois group acts on the set of lines on the correspondingcubic surface Y as the permutation (12)(34) in the notation of Remark 4.7. Thereforethere are seven lines defined over F q : E , E , L , L , L , Q and Q . Also there aretwo F q -points L ∩ L and L ∩ L , that are the points of intersection of the lines notdefined over F q .The line L meets each other line defined over F q , and any other line defined over F q meets only two lines defined over F q . Therefore there are 6 q + 3 points defined over F q on the lines E , E , L , L , Q and Q . Each of these lines does not intersect the lines L , L , L and L . Thus there are at least 6 q + 5 points defined over F q lying on thelines on Y .Note that there are q + 3 q + 1 points defined over F q on Y . If q q + 3 q + 1 q + 5, so either the surface Y does not exist over F , F and F , or itis impossible to blow up Y at an F q -point not lying on the lines. Therefore a del Pezzosurface of degree 2 of type 3 does not exist over F , F and F .By Remark 2.7, there are at most 7 q + 3 points defined over F q lying on the lines on Y . For q > q + 3 q + 1 > q + 3. Therefore one can blow up Y at an F q -pointnot lying on the lines and get a del Pezzo surface of degree 2 of type 3 by Corollary 2.6.To construct a cubic surface Y for q >
5, one can consider a smooth conic Q on P F q ,choose an F q -point and two points of degree 2 on Q , and blow up P F q at these five geometricpoints. On the obtained surface Z there are q + 2 q + 1 points defined over F q , and 4 q + 3points defined over F q lying on the lines. Therefore we can blow up Z at an F q -point notlying on the lines and get a required cubic surface Y by Corollary 2.6 and Lemma 2.9. (cid:3) Lemma 4.9.
A del Pezzo surface of degree of type does not exist over F and existsover other finite fields.Proof. If X has type 6 then the Galois group acts on the set of lines on the correspondingcubic surface Y as the permutation (12)(34)(56).Over any finite field F q the cubic surface Y can be constructed in the following way.Let p , p and p , p be two pairs of conjugate geometric points defined over F q in generalposition on P F q . Consider two smooth conjugate conics Q and Q defined over F q passingthrough p , p , p and p (such conics always exist since three reduced conics are definedover F q ). Let q be an F q -point on Q and q ∈ Q be the conjugate of q . We show thatthe points p i and q j are in general position.Any line passing through two points p i and p j does not contain other points lying on Q or Q . Therefore such line passes through exactly two points from the set { p i , q j } . Ifa point p i lie on a line L passing through q and q then the conjugate of p i lie on thisline since q and q are conjugate. Hence L meets Q at three points, that is impossible.Thus the points p i and q j are in general position. The blowup of Y → P F q at thesepoints is a required cubic surface by Theorem 2.2 and Lemma 2.9.There are three lines on Y defined over F q : L , L , and L . Note that the lines L , L , and L meet each other. Also there are six F q -points L ∩ L , L ∩ L , L ∩ L , L ∩ L , L ∩ L , and L ∩ L , that are the points of intersection of the lines notdefined over F q . Thus, by Remark 2.7, there are at most 3 q + 7 points defined over F q lying on the lines on Y . ote that there are q + q + 1 points defined over F q on Y . For q > q + q + 1 > q + 7. Therefore one can blow up Y at an F q -point not lying on thelines and get a del Pezzo surface of degree 2 of type 6 by Corollary 2.6.For q = 3 one can check that the seven points(1 : 0 : 1 + i) , (1 : 0 : 1 − i) , (0 : 1 : 1 + i) , (0 : 1 : 1 − i) , (1 : 1 + i : 0) , (1 : 1 − i : 0) (1 : − , are in general position.A del Pezzo surface of degree 2 of type 6 over F does not exist, since there are no seven F -points on P F in general position by [BFL16, Proposition 4.5]. (cid:3) Now we consider types 11, 14 and 23.
Lemma 4.10.
A del Pezzo surface of degree of type exists over all finite fields.Proof. A del Pezzo surface of type 11 is the blowup of a P F q at two points of degree 2 anda point of degree 3.Let r , q , q , q be geometric points on P F q in general position such that r is an F q -point, q i are conjugate points defined over F q . Consider two smooth conics Q and Q definedover F q and passing through these four points. Let L and L be two conjugate linesdefined over F q and passing through r , and p ij be the points of intersection of L i and Q j that differ from r . Then the points p ij are defined over F q . We show that the points p ij and q k are in general position.Any line passing through two points q i and q j does not contain other points lying on Q or Q . Therefore such line passing through exactly two points from the set { p ij , q k } .If a point q k lie on a line passing through two points from the set { p ij } then the points F q k and F q k lie on this line since F p ij = p ij for any i and j . But this is impossiblesince any line meets Q at 2 or 1 point. If a line L passing through three points from theset { p ij } then it has two common points with L or L . That is impossible.If a conic Q passes through the points q , q , q , and three points from the set { p ij } then it has five common points with Q or Q . That is impossible. If a conic Q passesthrough the six points p , p , p , p , q i , and q j , then the conics Q and F Q have atleast five common points. Therefore one has Q = F Q and this conic contains the points q , q , q and have five common points with Q and Q . That is impossible.Thus the points p ij and q k are in general position. The blowup of P F q at these pointsis a del Pezzo surface of degree 2 of type 11 by Theorem 2.2 and Lemma 2.9. (cid:3) Lemma 4.11.
A del Pezzo surface of degree of type does not exist over F and existsover other finite fields.Proof. A del Pezzo surface of type 14 is the blowup of a P F q at an F q -point, a point ofdegree 2 and a point of degree 4.Let us show that there are no a point of degree 2 and a point of degree 4 in generalposition on P F . Assume that there exist such points in general position and p , p , q , q , q , q are the corresponding geometric points. Then a conic Q i passing through p i , q , q , q , q is defined over F and not defined over F . There are only two conjugateconics defined over F passing through four points since there is only one point of degree 2 n P F . Let L ij be a line passing through q i and q j . Then the conics L ∪ L and L ∪ L are conjugate conics defined over F passing through q , q , q , q . Therefore Q and Q coincide with these conics, and points q and q lie on the union of lines L ij . Thus thepoints p , p , q , q , q , q are not in general position. We have a contradiction.Now assume that q >
3. Let q , q , q , and q be conjugate geometric points definedover F q in general position on P F q . Consider two smooth conjugate conics Q and Q defined over F q passing through q , q , q and q . Let p be an F q -point on Q and p ∈ Q be the conjugate of p . We show that the points p i and q j are in general position.Any line passing through two points q i and q j does not contain other points lying on Q or Q . Therefore such line passes through exactly two points from the set { p i , q j } . If apoint q i lie on a line passing through p and p then the points F q i , F q i and F q i lie onthis line since F p = p and F p = p . But this is impossible.Thus the points p i and q j are in general position. The blowup of Y → P F q at thesepoints is a cubic surface by Theorem 2.2. If we blow up an F q -point not lying on thelines on Y then we get a del Pezzo surface of degree 2 of type 14, by Corollary 2.6 andLemma 2.9. Such point exists since there are q + q + 1 points defined over F q on Y , andno more than q + 2 from those points lie on the lines (in the notation of Remark 4.7 thesepoints are F q -points on L and the F q -point L ∩ L ). (cid:3) Lemma 4.12.
A del Pezzo surface of degree of type exists over all finite fields.Proof. A del Pezzo surface of type 23 is the blowup of P F q at a point of degree 3 and apoint of degree 4.Let q , q , q be conjugate geometric points on P F q not lying on a line and definedover F q . There are q + q + 1 lines on P F q and q + q + 1 conics passing through q , q and q . Each of those curves contains q − q points of degree 4. There are q − q points ofdegree 4 on P F q . For any q > q − q > · q − q · ( q + q + 1) . Therefore we can find a point of degree 4 not lying on a line or a conic passing through q , q and q . Denote the corresponding geometric points by p , p , p and p . We showthat the points p i and q j are in general position.Assume that points p i , p j , and q k lie on a line. Then the points F p i = p i , F p j = p j ,and F q k lie on the same line. Therefore the points q , q , and q lie on a line. We havea contradiction. The same arguments show that three points p i , q j , and q k can not lie ona line.If a conic passes through six points p i , p j , p k , q , q , and q then this conic passesthrough the points p , p , p , p since the sets { F p i , F p j , F p k } and { p i , p j , p k } have twocommon points. If a conic passes through the six points p , p , p , p , q i , q j then itpasses through the points q , q , q since either F q i = q j or F q j = q i . All these cases areimpossible since the points p , p , p , p do not lie on a conic passing through the points q , q , and q .Thus the points p j and q j are in general position. The blowup of P F q at these points isa del Pezzo surface of degree 2 of type 23 by Theorem 2.2 and Lemma 2.9. (cid:3) he next five types of del Pezzo surface of degree 2 can be obtained by blowing upminimal del Pezzo surfaces of degree 4 of certain types at two F q -points. The method thatwe use for these types is very closely related to the method used in [Tr17, Lemma 3.12],where there is considered the blowup of the sixth type of a minimal del Pezzo surface ofdegree 4 at two F q -points. Lemma 4.13.
A del Pezzo surface of degree of type does not exist over F , F , F , F and exists over other finite fields.Proof. A del Pezzo surface of degree 2 of type 10 is the blowup of a minimal del Pezzosurface S of degree 4 at two F q -points. The surface S admits a conic bundle structurewith degenerate fibres over four F q -points. Such del Pezzo surface does not exist over F and F , and exists over other finite fields by [Ry05, Theorem 3.2] and [Tr17, Theorem 2.5].Assume that q >
4. The surface S admits two structures of conic bundles and each of the16 lines is a component of a singular fibre of one of these conic bundles. These lines formeight h F i -orbits, each consisting of 2 curves. Therefore there are eight F q -points on thelines, that are the points of intersection of conjugate lines. But by equation (1.1) thereare q − q + 1 points defined over F q on S . Thus for q > F q -point P on S not lying on the lines. Let f : e S → S be the blowup of S at P . By Corollary 2.6, thesurface e S is a cubic surface of type ( c ) (see Table 3). There are three lines on e S definedover F q : the exceptional divisor E = f − ( P ), and the proper transforms C and C of thefibres passing through P of the two conic bundles on S .Now we show that all other h F i -orbits of lines consist of 2 lines. Let H be a line on e S that differs from E , C and C . If H · E = 0 then f ( L ) is a ( − H · E = 1. Then C · H = C · H = 0 since E + C + C ∼ − K e S . It means that f ( H ) is a section of any conic bundle on S . For anysingular fibre this section must meet one component D of this fibre at a point, and forthe other component D of this fibre f ( H ) · D = 0. But we have F D = D , therefore F f ( H ) · D = f ( H ) · D = 1. Thus F f ( H ) = f ( H ) and the orbit of H consists of 2 lines.Therefore on e S there are twelve F q -points that are the points of intersection of conjugatelines, and three meeting each other lines E , C , and C defined over F q . By Remark 2.7,there are 3 q +12 − ǫ or 3 q +13 − ǫ points defined over F q lying on the lines on Y , where ǫ q − q + 1 points defined over F q on e S . For q > q − q + 1 > q + 13. Therefore one can blow up e S at an F q -point not lying on the linesand get a del Pezzo surface of degree 2 of type 10 by Corollary 2.6 and Lemma 2.9.By Lemma 2.8, for q = 5 there are at most 2 Eckardt points on a line. Therefore if E , C , and C meet at an Eckardt point then there are 3 q + 1 points defined over F q on theirunion and ǫ
3. In this case there are at least 25 F q -points lying on the lines on e S .Otherwise, there are 3 q points defined over F q on E ∪ C ∪ C and ǫ
6. In this case thereare at least 21 F q -points lying on the lines on e S . In the both cases there is no an F q -pointnot lying on the lines on e S .By Lemma 2.8, for q = 4 there are either 0, or 1, or 5 Eckardt points on a line. Thereforeif E , C and C meet at an Eckardt point then there are 3 q + 1 points defined over F q on their union and ǫ
12. In this case there are at least 13 F q -points lying on the lineson e S . Otherwise, there are 3 q points defined over F q on E ∪ C ∪ C and ǫ
3. In this ase there are at least 21 F q -points lying on the lines on e S . In the both cases there is noan F q -point not lying on the lines on e S .For q q − q + 1 < q . Therefore a del Pezzo surface of degree 2 of type 10can not be obtained over F , F , F , and F . (cid:3) Lemma 4.14.
A del Pezzo surface of degree of type exists over all finite fields.Proof. A del Pezzo surface of degree 2 of type 16 is the blowup of a minimal del Pezzosurface S of degree 4 at two F q -points. The surface S admits a conic bundle structurewith degenerate fibres over an F q -point and a point of degree 3. Such del Pezzo surfaceexists over all finite fields by [Ry05, Theorem 3.2]. The surface S admits two structuresof conic bundles and each of the 16 lines is a component of a singular fibre of one of theseconic bundles. The orbits of h F i on the set of the lines on S have cardinalities 2, 2, 6,and 6. Therefore there are two F q -points on the lines, that are the points of intersectionof the lines in the orbits of length two. But by equation (1.1) there are q + q + 1 pointsdefined over F q on S . Thus there is an F q -point P on S not lying on the lines. Let f : e S → S be the blowup of S at P . By Corollary 2.6, the surface e S is a cubic surface.There are three lines on e S defined over F q : the exceptional divisor E = f − ( P ), and theproper transforms C and C of the fibres passing through P of the two conic bundles.As in Lemma 4.13 one can check that the h F i -orbits of lines meeting E at a point havecardinalities 1, 1, 2, and 6.Therefore on e S there are three F q -points that are the points of intersection of pairs ofconjugate lines, and three meeting each other lines E , C , and C defined over F q . ByRemark 2.7, there are at most 3 q + 4 points defined over F q lying on the lines on Y .By equation (1.1), there are q + 2 q + 1 points defined over F q on e S . For q > q + 2 q + 1 > q + 4. Therefore one can blow up e S at an F q -point not lying on thelines and get a del Pezzo surface of degree 2 of type 10 by Corollary 2.6 and Lemma 2.9.For q = 2 we should more carefully choose a point P on S .Let π : S → P F and π : S → P F be the two conic bundles structures, s and s be F -points on a singular fibres of π and π respectively, F be a fibre of π containing s and F be a fibre of π containing s . If F and F have a common F -point P , then onecan blow up this point and get a cubic surface e S , such that at most eight F -points lyingon the lines on e S , since the two points of intersection of conjugate lines lie on lines definedover F .If F and F do not have a common F -point, then the other smooth fibre e F of π defined over F transversally intersects F at two F -points. One can blow up any of thesepoints and get a cubic surface e S , such that at most eight F -points lying on the lines on e S ,since one of the points of intersection of conjugate lines lies on a line defined over F andthe three lines defined over F do not have a common Eckardt point.In the both cases there are nine F -points on e S and one can blow up e S at an F -pointnot lying on the lines and get a del Pezzo surface of degree 2 of type 10. (cid:3) Now we consider del Pezzo surfaces of degree 2 of types 22, 29, and 30. These sur-faces are the blowups of del Pezzo surfaces of degree 4 with the Picard number 1 attwo F q -points. The blowup of such a surface at an F q -point not lying on the lines is a ubic surface admitting a structure of a conic bundle. These cubic surfaces exist overany field by [Ry05, Theorem 3.2], and have types ( c ), ( c ), and ( c ) (see Table 3) byLemma 2.9. Lemma 4.15.
A del Pezzo surface of degree of type does not exist over F and existsover other finite fields.Del Pezzo surfaces of degree of types and exist over all finite fields.Proof. From [Man74, Table 1] one can see, that the h F i -orbits of the lines of length atmost 3 have cardinalities 1, 2, 2, and 2 for a cubic surface of type ( c ), have cardinalities1 and 2 for a cubic surface of type ( c ), and have cardinality 1 for a cubic surface oftype ( c ). Therefore by Remark 2.7 there are at most q + 4, q + 2 and q + 1 points definedover F q and lying on the lines for cubic surfaces of types ( c ), ( c ) and ( c ) respectively.By equation (1.1), there are q − q + 1, q + q + 1 and q + 2 q + 1 points defined over F q on these types of cubic surfaces respectively.Note that for q > q − q + 1 > q + 4, and for q > q + q + 1 > q + 2,and q + 2 q + 1 > q + 1. Therefore one can blow up a cubic surface of type ( c ) for q > c ) or ( c ) over any finite field at an F q -point not lyingon the lines and get a del Pezzo surface of degree 2 of types 22, 29 or 30 respectively byCorollary 2.6 and Lemma 2.9.For q = 3 a cubic surface of type ( c ) has a structure of a conic bundle with threedegenerate fibres and one smooth fibre F defined over F . These singular fibres consistof pairs of conjugate lines defined over F , and the line defined over F is a bisectionof this conic bundle (see the proof of [Isk79, Theorem 4]). Therefore there are at leasttwo F -points on F not lying on the lines. One can blow up the cubic surface at one ofthese F -points and get a del Pezzo surface of degree 2 of type 22.For q = 2 a cubic surface of type ( c ) has a structure of a conic bundle with threedegenerate fibers defined over F . Therefore there is no an F -point not lying on the lines.Thus a del Pezzo surface of degree 2 of type 22 can not be obtained over F . (cid:3) The remaining case is a del Pezzo surface of degree 2 of type 38. This surface is theblowup of a minimal cubic surface of type ( c ) at an F q -point not lying on the lines. Lemma 4.16.
A del Pezzo surface of degree of type does not exist over F and existsover other finite fields.Proof. A del Pezzo surface of type 38 is the blowup of a cubic surface of type ( c )at an F q -point. From [Man74, Table 1] one can see that the h F i -orbits of lines havecardinalities 3, 6, 6, 6 and 6 for a cubic surface of type ( c ). Therefore there is at mostone F q -point lying on the lines for this cubic surface. By equation (1.1) there are q − q + 1points defined over F q on this cubic surface. One has q − q + 1 > q .For q > c ) exists by [RT17, Proposition 6.1]. Therefore onecan blow up this surface at an F q -point not lying on the lines and get a del Pezzo surfaceof degree 2 of type 38 by Corollary 2.6 and Lemma 2.9.For q = 2 a cubic surface of type ( c ) does not exist by [RT17, Proposition 6.5]. Thusa del Pezzo surface of degree 2 of type 38 can not be obtained over F . (cid:3) he blowups of minimal cubic surfaces at an F q -point of other types ( c )–( c ) areconsidered in [Tr17, Lemma 3.13]. But for the three types ( c )–( c ) of these surfacesthere are some restrictions on q coming from [RT17, Theorem 1.2]. In Section 5 we removethese restrictions for the types ( c ) and ( c ).Now we prove Theorem 1.2. Proof of Theorem 1.2.
Del Pezzo surfaces of degree 2 of types that are the Geiser twists(see Definition 4.2) of each other exist over the same finite fields. For each of the 30 pairsof types we give a reference, where one of these types is considered. • The types 1 and 49 are considered in [BFL16, Subsection 4.2, case a = 8]. • The types 2 and 31 are considered in [BFL16, Subsection 4.2, case a = 6]. • The types 3 and 18 are considered in Lemma 4.8. • The types 4 and 53 are considered in [BFL16, Subsection 4.2, case a = 5]. • The types 5 and 10 are considered in Lemma 4.13. • The types 6 and 9 are considered in Lemma 4.9. • The types 7 and 40 are considered in [Tr17, Proposition 2.17 and Lemma 3.5]. • The types 8 and 33 are considered in [BFL16, Subsection 4.2, case a = 4]. • The types 11 and 27 are considered in Lemma 4.10. • The types 12 and 55 are considered in [Tr17, Lemma 3.10]. • The types 13 and 22 are considered in Lemma 4.15. • The types 14 and 21 are considered in Lemma 4.11. • The types 15 and 54 are considered in [BFL16, Subsection 4.2, case a = 3]. • The types 16 and 19 are considered in Lemma 4.14. • The types 17 and 50 are considered in [Tr17, Lemma 3.12]. • The types 20 and 45 are considered in [Tr17, Proposition 2.17]. • The types 23 and 42 are considered in Lemma 4.12. • The types 24 and 43 are considered in [BFL16, Subsection 4.2, case a = 1]. • The types 25 and 38 are considered in Lemma 4.16. • The types 26 and 37 are considered in [BFL16, Subsection 4.2, case a = 2]. • The types 28 and 35 are considered in [Tr17, Proposition 2.17]. • The types 29 and 41 are considered in Lemma 4.15. • The types 30 and 34 are considered in Lemma 4.15. • The types 32 and 60 are considered in [Tr17, Lemma 3.13]. • The types 36 and 59 are considered in [Tr17, Lemma 3.11]. • The types 39 and 57 are considered in [Tr17, Lemma 3.10]. • The types 44 and 52 are considered in [Tr17, Proposition 2.17]. • The types 46 and 58 are considered in [Tr17, Lemma 3.13] and Lemma 5.5. • The types 47 and 56 are considered in [Tr17, Lemma 3.13] and [RT17]. • The types 48 and 51 are considered in [Tr17, Lemma 3.13] and Lemma 5.5. (cid:3) Del Pezzo surfaces of degree ⊂ W ( E ) we construct the correspondingcubic surface (that is a del Pezzo surface of degree 3) over F q if it is possible, and showthat such surfaces do not exist for other values of q . As a result we prove Theorem 1.3.In Table 3 we collect some facts about conjugacy classes of elements in the Weylgroup W ( E ). This table based on the tables in [SD67], [Man74], and [Car72]. The ype Graph Order Eigenvalues ρ ( X ) Γ Blowup c ∅ c A − −
1, 1, 1, 1, 1 5 3. c A − − − −
1, 1, 1 3 10. c D ( a ) 4 i, i, − i, − i, 1, 1 3 17. c A × A − i, − −
1, 1, 1 3 14. c A ω , ω , 1, 1, 1, 1 5 4. c D − ω , − ω , − −
1, 1, 1 3 16. c A × A ω , ω , − −
1, 1, 1 3 11. c A ω , ω , ω , ω , 1, 1 3 12. c A × A − ω , − ω , ω , ω , − − c A ω , ω , ω , ω , ω , ω c E ( a ) 6 − ω , − ω , − ω , − ω , ω , ω c E
12 i ω , i ω , − i ω , − i ω , ω , ω c E ( a ) 9 ξ , ξ , ξ , ξ , ξ , ξ c A ξ , ξ , ξ , ξ , 1, 1 3 15. c A −
1, 1, 1, 1, 1, 1 6 2. c A − − −
1, 1, 1, 1 4 6. c A − i, −
1, 1, 1, 1 4 8. c A × A − i, − − −
1, 1 2 22. c D ξ , ξ , ξ , ξ , −
1, 1 2 29. c A × A ω , ω , −
1, 1, 1, 1 4 7. c A × A ω , ω , ω , ω , −
1, 1 2 20. c A − ω , − ω , ω , ω , −
1, 1 2 26. c D ( a ) 12 − ω , − ω , i, − i, −
1, 1 2 30. c A × A ξ , ξ , ξ , ξ , −
1, 1 2 24.
Table 3.
Conjugacy classes of elements in W ( E )first column is a type according to [SD67]. The second column is the Carter graphcorresponding to the conjugacy class (see [Car72]). The third column is the order ofelement. The fourth column is the collection of eigenvalues of the action of element on K ⊥ X ⊂ Pic( X ) ⊗ Q . The fifth column is the invariant Picard number ρ ( X ) Γ . The lastcolumn is the type of the corresponding conjugacy class (see Table 2) in W ( E ) afterblowing up a cubic surface at an F q -point. Remark . Theorem 1.3 for cubic surfaces of types ( c ), ( c ), ( c ), ( c ), ( c ), ( c ), ( c )and ( c ) immediately follows from Theorem 1.2. One can consider a del Pezzo surface ofdegree 2 of the corresponding type (see the last column of Table 3) and contract a linedefined over F q .Many types of cubic surfaces are considered in [Ry05], [SD10], [RT17], [Tr17], [BFL16]and in some lemmas in Section 4. Actually, there are only three types of cubic surfacesleft: ( c ), ( c ), ( c ). These cubic surfaces are isomorhic to the blowup of P F q at a numberof points of certain degrees. We consider these types in the following three lemmas. Lemma 5.2.
A cubic surface of type ( c ) does not exist over F , F , and exists over otherfinite fields. roof. A cubic surface of type ( c ) is the blowup of P F q at two F q -points and two pointsof degree 2.In the proof of Lemma 4.8 it is shown that there are q + 3 q + 1 points defined over F q on X and at least 6 q + 5 points defined over F q lying on the lines on X . Therefore thissurface does not exist for q = 2 and q = 3, since q + 3 q + 1 < q + 5 in these cases.For q > X can be constructed in the following way. Let p , p and p , p be two pairs of conjugate geometric points defined over F q in general positionon P F q . Consider two smooth conics Q and Q defined over F q passing through p , p , p and p . Let q be an F q -point on Q and q be an F q -point on Q . We show that thepoints p i and q j are in general position.Any line passing through two points p i and p j does not contain other points lying on Q or Q . Therefore such line passes through exactly two points from the set { p i , q j } . If apoint p i lie on a line L passing through q and q then the conjugate of p i lie on this linesince q and q are F q -points. Hence L meets Q at three points, that is impossible.Thus the points p i and q j are in general position. The blowup of P F q at these points isa cubic surface of type ( c ) by Theorem 2.2 and Lemma 2.9. (cid:3) Lemma 5.3.
A cubic surface of type ( c ) does not exist over F and exists over otherfinite fields.Proof. A cubic surface of type ( c ) is the blowup of a P F q at two points of degree 3.Let q = 2 and P be a point on P F of degree 3 in general position. There are 7 smoothconics defined over F passing through P and 7 lines defined over F on P F . These linesand conics contain 22 points of degree three: 14 points on lines, 7 points that differsfrom P on conics passing through P , and P . But there are exactly 22 points of degree 3on P F . Therefore there are no two points of degree 3 in general position on P F .For q > c ) is constructed in the proof of [RT17, Proposi-tion 6.1]. (cid:3) Lemma 5.4.
A cubic surface of type ( c ) exists over all finite fields.Proof. A cubic surface of type ( c ) is the blowup of P F q at an F q -point, a point of degree 2and a point of degree 3.Consider a smooth conic Q on P F q . Let q and q be a pair of conjugate geometric pointson Q defined over F q , and q , q , q be three conjugate geometric points on Q definedover F q . Then any line L ij passing through two geometric points q i and q j except L does not contain F q -points.There are 2 q + 2 points defined over F q on Q ∪ L , and q + q + 1 points defined over F q on P F q . One has q + q + 1 > q + 2 for q >
2. Therefore there is an F q -point p on P F q not lying on L ij and Q .Thus the points p and q i are in general position. The blowup of P F q at these points isa cubic surface of type ( c ) by Theorem 2.2 and Lemma 2.9. (cid:3) For the three types ( c ), ( c ), and ( c ) of minimal cubic surfaces considered in [RT17]there are some restrictions on q . More precisely, cubic surfaces of types ( c ) and ( c )are constructed for any odd q , and cubic surfaces of type ( c ) are constructed for any q , uch that q = 6 k + 1. In the following lemma we construct cubic surfaces of types ( c )and ( c ) for even q . Also for even q this lemma give a construction of cubic surfaces oftype ( c ), that are considered in [SD10]. Lemma 5.5.
Cubic surfaces of types ( c ) and ( c ) exist over all finite fields.Proof. For any odd q cubic surfaces of types ( c ) and ( c ) are constructed in [RT17,Subsection 5.3] and [RT17, Subsection 5.2] respectively. We construct these types ofcubics surfaces for any even q .Let q be even and X be a cubic surface in P F q given by the equation A ( x z + xz + y + y z ) + Byz ( y + z ) + Cz + Dy ( y + z ) t + Ez t + F zt + t = 0 . One can check that the surface X is smooth if and only if A = 0.The automorphism g of order 4 given by g : ( x : y : z : t ) ( x + y : y + z : z : t )acts on X . The g -invariant hyperplane section z = 0 is the union of three lines Ay + Dy t + t = 0 . Moreover, one can select A and D so that these three lines are conjugate.Therefore the group Γ has order divisible by 3 and commute with the image e g of g inPic( X ). In particular, an element h ′ of order 3 in Γ commutes with e g . The element e gh ′ has order 12, thus the type of this element is either c , or c (see Table 3). Therefore theelement h ′ has type c or c . By [RT17, Lemma 2.10] an element of type c is conjugateto (123) ∈ S ⊂ W ( E ). Applying the notation of Remark 4.7 one can easily check thatthere are no (123)-orbits consisting of three meeting each other lines. Therefore h ′ hastype c and X is a minimal cubic surface of type ( c ), ( c ) or ( c ).Let h be a generator of Γ. Applying Proposition 4.1 one can obtain cubic surfaces X , X and X such that the corresponding groups Γ , Γ and Γ in W ( E ) are generated by e gh , e g h and e g h respectively. We want to show that there are a surface of type ( c ), asurface of type ( c ) and two surfaces of type ( c ) among the four surfaces X , X , X and X .If X has type ( c ) or ( c ) then the surface X has type ( c ), so for these two caseswe can replace X by X and reduce these cases to the case when X has type ( c ).Assume that X has type ( c ). The elements h and e g can be simultaneously diagonalizedin GL (cid:0) K ⊥ X ⊗ Q (cid:1) , and e g multiplies the eigenvalues of h by i, − i, i, − i, 1 and 1. Theelement e g h has order 12 and type c , therefore e g trivially acts on the eigenvalues ω and ω of h . Thus all eigenvalues of the element e gh are ω or − ω , and this element has type c or c . For these cases the element e g h has type c or c respectively. Hence two of thesurfaces X , X , X , and X have type ( c ), one surface has type ( c ) and one surfacehas type ( c ). (cid:3) Now we prove Theorem 1.3.
Proof of Theorem 1.3.
For each type of cubic surfaces we give a reference, where this typeis considered. • The type ( c ) is considered in [SD10] (see also [BFL16, Corollary 3.3]). • The type ( c ) is considered in Lemma 5.2. The type ( c ) is considered in the proof of Lemma 4.13. • The type ( c ) is considered in the proof of [Tr17, Lemma 3.12]. • The type ( c ) is considered in the proof of Lemma 4.11. • The type ( c ) is considered in [BFL16, Subsection 3.1, case a = 4]. • The type ( c ) is considered in Lemma 5.3. • The type ( c ) is considered in [RT17, Propositions 6.1 and 6.5] • The type ( c ) is considered in [SD10]. • The type ( c ) is considered in Lemma 5.5. • The type ( c ) is considered in Lemma 5.5. • The type ( c ) is considered in [RT17, Subsection 5.4]. • The type ( c ) is considered in [SD10] (see also [BFL16, Subsection 3.1, a = 5]). • The type ( c ) is considered in the proof of Lemma 4.9. • The type ( c ) is considered in [BFL16, Subsection 3.1, case a = 3]. • The type ( c ) is considered in [Ry05, Theorem 3.2]. • The type ( c ) is considered in Lemma 5.4.The types ( c ), ( c ), ( c ), ( c ), ( c ), ( c ), ( c ) and ( c ) are considered in Remark 5.1. (cid:3) Del Pezzo surfaces of degree ⊂ W ( D ) we construct the corresponding delPezzo surface of degree 4 over F q if it is possible, and show that such surfaces do not existfor other values of q . As a result we prove Theorem 1.4.In Table 4 we collect some facts about conjugacy classes of elements in the Weylgroup W ( D ). The Weyl group W ( D ) is isomorphic to ( Z / Z ) ⋊ S ⊂ GL ( Z ). Theelements of ( Z / Z ) change signes of even number of the coordinates, and the elementsof S permute the coordinates (see [DI09, Subsection 6.4]). One can find the classificationof conjugacy classes in ( Z / Z ) ⋊ S in [DI09, Table 3], but there are two missed classes(denoted by ( ab ) ι ac and ( ab ) ι acde in Table 4). We give a corrected classification in Table 4and prove that there are no missed classed in Proposition 6.1.The first column is a conjugacy class in the notation of [DI09, Subsection 6.4]. Thesecond column is the Carter graph corresponding to the conjugacy class (see [Car72]). Thethird column is the order of element. The fourth column is the collection of eigenvaluesof the action of element on K ⊥ X ⊂ Pic( X ) ⊗ Q . The fifth column is the invariant Picardnumber ρ ( X ) Γ . The last column is the type of the corresponding conjugacy class (seeTable 3) of W ( E ) after blowing up a del Pezzo surface of degree 4 at an F q -point. Proposition 6.1.
All conjugacy classes of ( Z / Z ) ⋊ S are listed in Table 4.Proof. Let g be an element in ( Z / Z ) ⋊ S . The image of g under the natural ho-momorphism f : ( Z / Z ) ⋊ S → S belongs to one of the seven conjugacy classes in S parametrised by Young diagrams: trivial, (12), (123), (12)(34), (1234), (12345), (123)(45).If f ( g ) is trivial then g is trivial or S -conjugate to ι or ι . These three cases arelisted in Table 4.If f ( g ) is conjugate to (12) then g is S -conjugate to (12), (12) ι , (12) ι , (12) ι ,(12) ι or (12) ι One has ι (12) ι ι = (12) and ι (12) ι ι = (12) ι . The otherfour cases are listed in Table 4. lass Graph Order Eigenvalues ρ ( X ) Γ Blowupid ∅ c ( ab ) A −
1, 1, 1, 1, 1 5 c ( ab )( cd ) A − −
1, 1, 1, 1 4 c ι ab A − −
1, 1, 1, 1 4 c ( abc ) A ω , ω , 1, 1, 1 4 c ( ab ) ι cd A − − −
1, 1, 1 3 c ( abcd ) A − i, −
1, 1, 1 3 c ( ab ) ι ac A − i, −
1, 1, 1 3 c ( abc )( de ) A × A ω , ω , −
1, 1, 1 3 c ι abcd A − − − −
1, 1 2 c ( ab )( cd ) ι ae A × A − i, − −
1, 1 2 c ( ab )( cd ) ι ac D ( a ) 4 i, i, − i, − i, 1 2 c ( abcde ) A ξ , ξ , ξ , ξ , 1 2 c ( abc ) ι de A × A ω , ω , − −
1, 1 2 c ( abc ) ι ad D − ω , − ω , − −
1, 1 2 c ( ab ) ι acde A × A − i, − − − c ( abcd ) ι ae D ξ , ξ , ξ , ξ , − c ( abc )( de ) ι ad D ( a ) 12 − ω , − ω , i, − i, − c Table 4.
Conjugacy classes of elements in W ( D )If f ( g ) is conjugate to (123) then g is S -conjugate to (123), (123) ι , (123) ι , (123) ι ,(123) ι or (123) ι . One has ι (123) ι ι = (123), ι (123) ι ι = (123) ι and ι (123) ι ι = (123) ι . The other three cases are listed in Table 4.If f ( g ) is conjugate to (12)(34) then g is S -conjugate to (12)(34), (12)(34) ι ,(12)(34) ι , (12)(34) ι , (12)(34) ι or (12)(34) ι One has ι (12)(34) ι ι = (12)(34), ι (12)(34) ι ι = (12)(34) and ι (12)(34) ι ι = (12)(34) ι . The other three casesare listed in Table 4.If f ( g ) is conjugate to (1234) then g is S -conjugate to (1234), (1234) ι ,(1234) ι , (1234) ι , (1234) ι or (1234) ι . One has ι (1234) ι ι = (1234), ι (1234) ι ι = (1234), ι (1234) ι ι = (1234) and ι (1234) ι ι = (1234) ι . Theother two cases are listed in Table 4.If f ( g ) is conjugate to (12345) then g is S -conjugate to (12345), (12345) ι , (12345) ι or (12345) ι . One has ι (12345) ι ι = (12345), ι (12345) ι ι = (12345)and ι (12345) ι ι = (12345). The remaining case is listed in Table 4.If f ( g ) is conjugate to (123)(45) then g is S -conjugate to (123)(45),(123)(45) ι , (123)(45) ι , (123)(45) ι , (123)(45) ι or (123)(45) ι . Onehas ι (123)(45) ι ι = (123)(45), ι (123)(45) ι ι = (123) ι , ι (123)(45) ι ι = (123)(45) ι and ι (123)(45) ι ι = (123)(45). The othertwo cases are listed in Table 4. (cid:3) emark . The result of Theorem 1.4 for del Pezzo surfaces of degree 4 of types, thatdiffer from id, ( ab )( cd ), ι ab , ι abcd , ( ab )( cd ) ι ae , ( ab )( cd ) ι ac , immediately follows from Theo-rem 1.3. One can consider a del Pezzo surface of degree 3 of the corresponding type (seethe last column of Table 4) and contract a line defined over F q .The type id is considered in [BFL16, Lemma 3.1], and the types ι abcd and ( ab )( cd ) ι ac ofdel Pezzo surfaces of degree 4 are considered in [Ry05] and [Tr17]. The type ( ab )( cd ) isisomorphic to the blowup of P F q at an F q -point and two points of degree 2. We considerthis type in the following lemma. Lemma 6.3.
A del Pezzo surface of degree of type ( ab )( cd ) does not exist over F andexists over other finite fields.Proof. A del Pezzo surface of degree 4 of type ( ab )( cd ) is the blowup of a P F q at an F q -pointand two points of degree 2. Five geometric points on P F q are in general position if andonly if they lie on a smooth conic by [BFL16, Lemma 2.4].There are no an F q -point and two points of degree 2 on a smooth conic Q in P F q for q = 2. For other values of q one can blow up points of these degrees on Q and get adel Pezzo surface of degree 4 of type ( ab )( cd ) by Theorem 2.2 and Lemma 2.9. (cid:3) The two remaining cases are the blowups of a quadric surface in P F q at four geometricpoints. This surface is isomorphic to P F q × P F q over F q . We denote by π and π the projec-tions on the first and the second factors of P F q × P F q . The Picard group Pic (cid:16) P F q × P F q (cid:17) isgenerated by the classes F and F of fibres of π and π respectively. Thus any divisor D on P F q × P F q is linearly equivalent to aF + bF . The pair ( a, b ) is called bedegree of D .For example the anticanonical class − K P F q × P F q has bedegree (2 , P F q × P F q . Proposition 6.4.
Let d , and p , . . . , p − d be − d points on P k × P k such that • no two lie on a fibre of π or π ; • no four lie on a curve of bedegree (1 , ; • no six lie on a curve of bedegree (1 , or (2 , ; • for d = 1 the points are not on a singular curve of bedegree (2 , with singularityat one of these points.Then the blowup of P k × P k at p , . . . , p − d is a del Pezzo surface of degree d .Moreover, any del Pezzo surface X of degree d over algebraically closed field k is the blowup of such set of points. Definition 6.5.
As in Definition 2.3 we say that a collection of points on a smoothquadric Q ⊂ P F k is in general position if it satisfies the conditions of Proposition 6.4.Now we can consider the cases ι ab and ( ab )( cd ) ι ae of Table 4. Lemma 6.6.
A del Pezzo surface of degree of type ι ab does not exist over F , F , andexists over other finite fields.Proof. A del Pezzo surface of degree 4 of type ι ab is the blowup of P F q × P F q at two pointsof degree 2. or q = 2 there are only two geometric fibres of π defined over F and not definedover F . Therefore one can not find two points of degree 2 in general position for thiscase.For q = 3 there are 18 points of degree 2 not lying in one geometric fibre of π or π on P F × P F , that are the points of degree 2 in general position. Let P be one of thesepoints. Then the union of the two geometric fibres of π and the two geomertic fibresof π passing through the two geometric points corresponding to P contains 10 pointsof degree 2 in general position (including P ). There are four curves of bedegree (1 , F and passing through P . These curves contain 8 points of degree 2 ingeneral position that differ from P . Therefore any point of degree 2 in general position,either lies on one of these curves, or lies in the geometric fibres of π or π containing P .Thus there are no pairs of points of degree 2 in general position on P F × P F .For q > c ) (this surface is constructed inLemma 5.2), contract a line defined over F q and get a del Pezzo surface of degree 4 oftype ι ab . (cid:3) Lemma 6.7.
A del Pezzo surface of degree of type ( ab )( cd ) ι ae does not exist over F and exists over other finite fields.Proof. A del Pezzo surface of degree 4 of type ( ab )( cd ) ι ae is the blowup of a conic Q ⊂ P F q ,such that ρ ( Q ) = 1 at a point of degree 4.For q = 2 there are five F -points and 66 points of degree 4 on Q . There are fivepairs defined over F conjugate fibres of π and π each containing 6 points of degree 4,ten curves of bedegree (1 ,
1) passing through three F -points each containing 3 points ofdegree 4, and six points of degree 4, that are the intersection of two geometric fibres F and F of π and two geometric fibres F and F of π such that the fibres F , F , F and F are permuted by the Galois group Gal ( F / F ). One has 5 · · Q for q = 2.For q > c ) (this surface is constructed inthe proof of Lemma 4.11), contract a line defined over F q and get a del Pezzo surface ofdegree 4 of type ( ab )( cd ) ι ae . (cid:3) Now we prove Theorem 1.4.
Proof of Theorem 1.4.
For each type of del Pezzo surfaces of degree 4 we give a reference,where this type is considered. • The type id is considered in [BFL16, Lemma 3.1]. • The type ( ab )( cd ) is considered in Lemma 6.3. • The type ι ab is considered in Lemma 6.6. • The type ι abcd is considered in [Ry05, Theorem 3.2] and [Tr17, Theorem 2.5]. • The type ( ab )( cd ) ι ae is considered in Lemma 6.7. • The type ( ab )( cd ) ι ac is considered in [Ry05, Theorem 3.2].The other types are considered in Remark 6.2. (cid:3) . Open questions
In this section we discuss open questions that arise for constructing del Pezzo surfacesof degree 1 over finite fields.By [Car72, Table 11] there are 112 types of del Pezzo surfaces of degree 1. This numberis greater than the sum of the numbers of del Pezzo surfaces types of degrees 2, 3 and 4.For 30 types of del Pezzo surfaces X of degree 1 one has ρ ( X ) = 1.As in the case of del Pezzo surfaces of degree 2 one can define the Bertini twist , sinceeach del Pezzo surface X of degree 1 has an involution defined by the double coverof P F q (1 : 1 : 2) given by the linear system | − K X | . But the Bertini twist X of a delPezzo surface X of degree 1 such that ρ ( X ) = 1 can have ρ ( X ) = 1. Actually, for fivepairs of types of del Pezzo surfaces X and X of degree 1 one has ρ ( X ) = ρ ( X ) = 1.Moreover, there are seven types of del Pezzo surfaces X of degree 1 with ρ ( X ) = 1, suchthat the Bertini twist of X has the same type as X .For nonminimal del Pezzo surfaces of degree 1 the methods that works for del Pezzosurfaces of degree at least 2 does not work. To apply Theorem 2.2 one has to check thateight geometric points of the blowup do not lie on a cubic curve having a singularity atone of these points. Also to apply Corollary 2.6 to del Pezzo surfaces X of degree 2 onehas to consider an additional condition, that the point of the blowup do not lie on theramification divisor of the anticanonical map X → P F q . For even q there arise additionaldifficulties (see [BFL16, Lemma 4.1]). Moreover, on del Pezzo surfaces of degree 2 a pairof lines permutted by the Geiser involution can have two or one common geometric points,and there can be four lines meeting each other either in six distinct geometric points, orin one common geometric point that is called a generalised Eckardt point . Therefore fordel Pezzo surfaces of degree 2 formulas for calculating F q -points lying on the lines aremuch more complicated than the formulas given in Remark 2.7.To construct some types of del Pezzo surfaces of degree 1 that admit a structure of aminimal conic bundle one can apply [Ry05, Theorem 2.11]. But a constructed minimalsurface admitting a structure of a conic bundle is not in general a del Pezzo surface,as in the cases of degrees 4 and 2 (see the proof of [Ry05, Theorem 3.2] and [Tr17,Proposition 2.3] respectively).Despite these problems some types of del Pezzo surfaces of degree 1 are constructed in[BFL16, Section 5] and [Tr17, Lemma 3.11].A generalization of the results of this paper to the case of del Pezzo surfaces of degree 1is a great challenge. References [BFL16] B. Banwait, F. Fit´e, D. Loughran, Del Pezzo surfaces over finite fields and their Frobenius traces,preprint, see http://arxiv.org/abs/1606.00300[Car72] R. W. Carter, Conjugacy classes in the weyl group, Compositio Mathematica, Vol. 25, Fasc. 1,1972, 1–59[DI09] I. V. Dolgachev, V. A. Iskovskikh, Finite subgroups of the plane Cremona group, In: Algebra,arithmetic, and geometry, vol. I: In Honor of Yu. I. Manin, Progr. Math., 269, 443–548, Birkh¨auser,Basel, 2009[DD17] I. Dolgachev, A. Duncan, Automorphisms of cubic surfaces in positive characteristic, preprint,see http://arxiv.org/abs/1712.01167[Fr51] J. S. Frame, The classes and representations of the groups of 27 lines and 28 bitangents, Annali diMatematica Pura ed Appl. Ser. IV, 32, 1951, 83–119 Fr67] J. S. Frame, The characters of the Weyl group E , in: Computational Problems in Abstract Alge-bra, ed. J. Leech, Oxford, 1967, 111–130[Isk79] V. A. Iskovskikh, Minimal models of rational surfaces over arbitrary field, Math. USSR Izv., 1979,43, 19–43 (in Russian)[Kap13] N. Kaplan, Rational point counts for del Pezzo surfaces over finite fields and coding theory,Harvard Ph.D. Thesis.[KR16] A. Knecht, K. Reyes, Full Degree Two del Pezzo Surfaces over Small Finite Fields, In: Contem-porary Developments in Finite Fields and Applications, 2016, 145–159[Man74] Yu. I. Manin, Cubic forms: algebra, geometry, arithmetic, In: North-Holland MathematicalLibrary, Vol. 4, North-Holland Publishing Co., Amsterdam-London; American Elsevier PublishingCo., New York, 1974[RT17] S. Rybakov, A. Trepalin, Minimal cubic surfaces over finite fields, Mat. Sb., 208:9, 148–170, 2017.Engl. transl.: Sb. Math., 208:9 (2017)[Ry05] S. Rybakov, Zeta-functions of conic bundles and del Pezzo surfaces of degree 4 over finite fields,Moscow Math. Journal volume 5:4, 2005, 919–926[SD67] H. P. F. Swinnerton-Dyer, The zeta function of a cubic surface over a finite field, Proceedings ofthe Cambridge Philosophical Society, 63, 1967, 55–71[SD10] H. P. F. Swinnerton-Dyer, Cubic surfaces over finite fields, Math. Proceedings of the CambridgePhilosophical Society, 2010, 149, 385–388.[Tr17] A. Trepalin, Minimal del Pezzo surfaces of degree 2 over finite fields, Bull. Korean Math. Soc.,54:5, 1779–1801, 2017[Ur96] T. Urabe, Calculation of Manin’s invariant for del Pezzo surfaces, Math. Comp., 65:213, 247–258,S15–S23, 1996[VNT18] S. Vlˇadut¸, D. Nogin, M. Tsfasman, Varieties over finite fields: quantitative theory, to appear inUspekhi Mat. Nauk Andrey Trepalin
Institute for Information Transmission Problems, 19 Bolshoy Karetnyi side-str., Moscow 127994, Russia [email protected]@mccme.ru