Zeta-functions of conic bundles and of del Pezzo surfaces of degree 4 over finite fields

Rybakov, Sergey

doi:10.1070/rm2005v060n05abeh003737

Cited by 5 publications

(6 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Proof. Part (1) follows from the fundamental exact sequence from class field theory for F q (t); see [29,Cor. 2.10].…”

Section: Del Pezzo Surfacesmentioning

confidence: 99%

“…The blow down of these lines yields the required surface. For a = −2, 0, the required surfaces have been constructed by Rybakov [29,Thm. 3.2] (these are X and XV III from Table 7.1, respectively).…”

Section: 2mentioning

confidence: 99%

“…Trace 0 can be obtained by blowing-up certain collections of closed points of total degree 7, and contracting a line. The existence of the remaining traces −1, −2 can be deduced from work of Rybakov [29] and Swinnerton-Dyer [37], respectively. 1.2. Corrections to Manin's and Swinnerton-Dyer's tables.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Del Pezzo surfaces over finite fields and their Frobenius traces

S.¹,

Fité²,

Loughran³

2018

Math. Proc. Camb. Phil. Soc.

View full text Add to dashboard Cite

Let S be a smooth cubic surface over a finite field F q . It is known that #S(F q ) = 1 + aq + q 2 for some a ∈ {−2, −1, 0, 1, 2, 3, 4, 5, 7}. Serre has asked which values of a can arise for a given q. Building on special cases treated by Swinnerton-Dyer, we give a complete answer to this question. We also answer the analogous question for other del Pezzo surfaces, and consider the inverse Galois problem for del Pezzo surfaces over finite fields. Finally we give a corrected version of Manin's and Swinnerton-Dyer's tables on cubic surfaces over finite fields.Note that a smooth cubic surface S with a(S) = 7 is split, i.e. all its lines are defined over F q . It has been known for a long time, prior to the work of Swinnerton-Dyer [37], that such a surface exists over F q if and only if q = 2, 3, 5; this result appears to be first due to Hirschfeld [16, Thm. 20.1.7].We briefly explain the proof of Theorem 1.1. Any smooth cubic surface over an algebraically closed field is the blow-up of P 2 in 6 rational points in general position. Whilst this does not hold over other fields in general, the trace values 1, 2, 3, 4, 5 can be obtained from cubic surfaces which are blow-ups of P 2 in collections of closed points in general position. We therefore show that such collections exist over every finite field F q , via combinatorial arguments. Trace 0 can be obtained by blowing-up certain collections of closed points of total degree 7, and contracting a line. The existence of the remaining traces −1, −2 can be deduced from work of Rybakov [29] and Swinnerton-Dyer [37], respectively. 1.2. Corrections to Manin's and Swinnerton-Dyer's tables. Let S be a smooth cubic surface over a finite field F q . Building on work of Frame [12] and Swinnerton-Dyer [36], Manin constucted a table (Table 1 of [27, p. 176]) of the conjugacy classes of W (E 6 ) and their properties, such as the trace a(S).Urabe [39,40] was the first to notice that Manin's table contains mistakes regarding the calculation of H 1 (F q , PicS) (the issue being that H 1 (F q , PicS) must have square order). In our investigation we found some new mistakes. These concern the Galois orbit on the lines, where the mistake can be traced back to , and the index [27, §28.2] of the surface. The index is the size of the largest Galois invariant collection of pairwise skew lines overF q .Manin's table has a surface of index 2, which led him to state [27, Thm. 28.5(i)] that the index can only take one of the values 0, 1, 2, 3, 6. Our investigations reveal, however, that index 2 does not occur, and that index 5 can occur, hence the correct statement is the following. Theorem 1.2. Let S be a smooth cubic surface over a finite field. Then the index of S can only take one of the values 0, 1, 3, 5, 6.

show abstract

“…Proof. Part (1) follows from the fundamental exact sequence from class field theory for F q (t); see [29,Cor. 2.10].…”

Section: Del Pezzo Surfacesmentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

See 1 more Smart Citation

Del Pezzo surfaces over finite fields and their Frobenius traces

S.¹,

Fité²,

Loughran³

2018

Math. Proc. Camb. Phil. Soc.

View full text Add to dashboard Cite

show abstract

“…In this case Y is either rational or a minimal del Pezzo surface of degree 4. In [Ry05] the first author constructs all types of minimal del Pezzo surfaces of degree 4 for q > 3. In this paper we explicitly construct minimal cubic surfaces with all possible zeta functions over many finite fields.…”

Section: Introductionmentioning

confidence: 99%

“…Del Pezzo surfaces of degree greater than 3 are birationally isomorphic to conic bundles. This observation allowed the first author to construct minimal del Pezzo surfaces of degree 4 with a given zeta function in [Ry05]. Minimal cubic surfaces are not birational to conic bundles, and one has to find another way to construct them.…”

Section: Introductionmentioning

confidence: 99%

Minimal cubic surfaces over finite fields

Rybakov¹,

Trepalin²

2017

Sb. Math.

View full text Add to dashboard Cite

Let $X$ be a minimal cubic surface over a finite field $\mathbb{F}_q$. The image $\Gamma$ of the Galois group $\operatorname{Gal}(\overline{\mathbb{F}}_q / \mathbb{F}_q)$ in the group $\operatorname{Aut}(\operatorname{Pic}(\overline{X}))$ is a cyclic subgroup of the Weyl group $W(E_6)$. There are $25$ conjugacy classes of cyclic subgroups in $W(E_6)$, and $5$ of them correspond to minimal cubic surfaces. It is natural to ask which conjugacy classes come from minimal cubic surfaces over a given finite field. In this paper we give a partial answer to this question and present many explicit examples.Comment: Final version, published in Math. Sb. 20 pages, 1 tabl

show abstract