Del Pezzo surfaces over finite fields and their Frobenius traces

Abstract

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.

Locally Recoverable Codes from Algebraic Curves and Surfaces

“…, f t } form a basis for L(t · ∞). We take e 1 = 1 and e 2 = w, where w satisfies the polynomial (4). Note that the degree of e 2 is 3.…”

Potentially Stably Rational Del Pezzo Surfaces over Nonclosed Fields

“…For n = 3, 2, and 1, the Galois group G k of k acts on the primitive Picard group of S (the orthogonal complement of the canonical class in Pic(S)) through the Weyl group W (E 9−n ); for n = 4 and conic bundles with n + 1 degenerate fibers through W (D n+1 ). These actions have been extensively studied, in connection with arithmetic applications and rationality questions, e.g., the Hasse Principle and Weak Approximation, when k is a number field (see e.g., [Man67], [KST89], [SD67], [Ura96], [Li], [BFL16]). This note is inspired by a recent result of Colliot-Thélène concerning stable rationality of geometrically rational surfaces over quasi-finite k, i.e., perfect fields with procyclic absolute Galois groups [CT17].…”

Planar arcs

“…The next question to answer ("Il primo nuovo quesito da porsi" as Segre writes in his 1955 paper) is whether there exists a complete arc of size q − 1 in PG(2, q). Since the standard frame extended with the points (1, 2, 3) and (1,3,4) gives a complete arc of size 6 for q = 7, the answer is affirmative. However, in spite of this small example, Segre expected the answer to be negative for sufficiently large q: "Rimane tuttavia da indagare se, com'è da ritenersi probabile, la risposta alla suddetta domanda non diventi invece negativa quando si supponga q sufficientemente grande".…”

“…The matrix whose columns are vector representatives of the points of an arc, generates a linear maximum distance separable code, see [18,Chapter 11] for more details on this. Other areas in which planar arcs play a role include the representation of matroids, see [20], Del Pezzo surfaces over finite fields, see [3], bent functions, see [19,Chapter 7] and pro-solvable groups, see [11].…”

“…The planar arc of 10 points in PG(2, 11), A = {(1, 0, 0), (0, 1, 0), (0, 0, 1),(1,8,7),(1,5,2),(1,3,8),(1,4,1)…”