The aim of this article is to construct codes from algebraic surfaces. Codes coming from algebraic geometry have seen a growing interest since their original contruction by Goppa. We consider evaluation codes: given an algebraic variety defined over a finite field , and an effective divisor on it, we evaluate the global sections in at the rational points of . We get a code of length , the number of rational points, and of dimension if the evaluation map is injective. It remains to determine the minimum distance.
Most of the algebraic geometry codes studied in the literature are based on algebraic curves, since Riemann-Roch theorem for curves is a powerful tool to estimate the dimension of the space of functions with given poles, and the number of zeroes of such a function never exceeds its number of poles. The study of evaluation codes over higher dimensional varieties is more difficult. On one hand, Riemann-Roch theorem involves the dimensions of the higher cohomology spaces. On the other hand, one has to bound the maximal number of rational points of the schemes defined by global sections of in order to estimate the minimum distance. Such a scheme is no longer zero-dimensional and might be singular or reducible. As a consequence one has to use new tools for the estimation of its number of rational points.
In the case of surfaces, one first has to determine the intersection pairing. Then, Riemann-Roch theorem often gives a lower bound for the dimension of the space of global sections. We want to use adjunction formula, to get the arithmetic genus of the curves coming from the zeroes of such a section, then to apply the Hasse-Weil-Serre bound on the number of points of irreducible curves. In order to do this, one also has to control their decomposition into irreducible components.
This can be done working in the Picard lattice of the surface . In this group, a decomposition of the zero scheme of a global section of corresponds to writing the class of as a sum of classes of effective divisors. A very efficient way to control this is to ask the Picard lattice to have rank [Zar07, Lemma 2.1].
We shall work on del Pezzo surfaces. A del Pezzo surface of degree over an algebraically closed field is obtained from the projective plane by blowing up points in general position (see Section 3). As a consequence, such surfaces are close to the plane; the arithmetic and geometric theory of these objects has been thoroughly studied (the classical reference is [Man86]) and we can explicit all invariants above: the intersection pairing and the (geometric) Picard lattice have fairly simple descriptions. They come endowed with an action of the Frobenius automorphism, and the (arithmetic) Picard lattice is the sublattice fixed by this action.
1.1. Our contribution
We construct del Pezzo surfaces of degree , and having Picard rank . In each case the Picard lattice is generated by the canonical divisor , and the cone of effective divisors by the anticanonical divisor . We call the code associated to this last divisor anticanonical; it has dimension (except for very small values of ), and all zero schemes of global sections are irreducible with arithmetic genus from the adjunction formula. As a consequence, they have at most points from the Hasse-Weil-Serre bound, and this provides very good codes. Some of them turn out to beat the best known codes listed in the database codetables [Gra07].
A central role is played by the Frobenius action on the Picard lattice of the surface, i.e. by the conjugacy class of the image of the Frobenius in a certain Weyl group. It gives many properties of the surface, such as the Picard rank or the number of rational points. Moreover, with the help of Galois cohomology, it enables us to determine the -rational automorphisms of the surface; since an automorphism of the surface must preserve the (anti)canonical class, we deduce some automorphisms of the codes.
Along the article, we try to be as constructive as possible. We give explicit descriptions of the anticanonical models. We also use birational morphisms from our surfaces to the projective plane to give explicit constructions of the codes, and Cremona transformations to describe the automorphisms.
1.2. Related works
In some sense, this paper fills a gap in the study of algebraic geometric codes constructed from del Pezzo surfaces (even if most of the authors cited below do not mention the fact that they work on del Pezzo surfaces). When is the projective plane (the only del Pezzo surface of degree over a finite field), the Picard group is generated by the class of a line, and the evaluation code associated to is the well-known projective Reed-Muller code of order (see [Lac88]). There are two types of del Pezzo surfaces of degree , one having Picard rank (the hyperbolic quadric, isomorphic to ), and the other having Picard rank (the elliptic quadric). Codes over these surfaces have been studied by Edoukou [Edo08], and Couvreur and Duursma [CD13]. In [Cou11, Section 3.2], the second author constructs good codes coming from del Pezzo surfaces of degree having Picard rank . In [LS18], Little and Schenck consider anticanonical codes on del Pezzo surfaces of degree and having Picard rank . Boguslavsky [Bog98] gives the parameters of anticanonical codes on split del Pezzo surfaces, i.e. on surfaces having maximal Picard rank.
Let us finally mention some works on codes on other blowups of the plane [Dav11, Bal13]. These blowups are no longer del Pezzo surfaces: since the blown up points lie on one or two lines, they are not in general position. Moreover the evaluation set is not the set of rational points of the surface, but the torus .
1.3. Outline of the article
The paper is organized as follows. In Section 3, we recall the necessary material on del Pezzo surfaces, and give the classification of such surfaces of degrees and over a finite field. Then we construct degree del Pezzo surfaces in Section 4; we give the parameters of the associated anticanonical codes, and determine the automorphisms of the surfaces. We construct del Pezzo surfaces of degree and the corresponding codes in Section 5. We give two geometric constructions of these codes: the first one by evaluating linear forms at the rational points of a surface embedded in and the second one by evaluating particular quintics forms at some rational points of the projective plane. Then, we determine the automorphism group of the surface. Finally, we construct del Pezzo surfaces of degree in Section 6, and determine the parameters of the associated codes.
The authors would like to thank Markus Grassl for pointing out the existence of automorphisms of the codes.
2. Context and notation
In the following, we fix a finite field of characteristic and an algebraic closure . We denote by a generator of the absolute Galois group . The projective space of dimension over is denoted by . On a surface , the intersection product of two divisor classes is denoted by and the self intersection of is denoted by . Given an effective divisor , the complete linear system associated to is denoted by , the Picard group of is denoted by and the canonical class of is denoted by .
Given a smooth projective geometrically connected surface over and divisor on , the code is defined as the image of the map
Note that for the map to be well–defined, one needs to order the rational points of , which can be done arbitrarily since the choice of another order would provide an isometric code with respect to the Hamming metric. Similarly, the evaluation of a global section of at a point depends on the choice of a generator of the stalk of the sheaf at but choosing another system of generators would provide an isometric code. Since we are mostly interested in the parameters of the code : its dimension and minimum distance, it is sufficient to consider our code up to isometry.
3. del Pezzo surfaces
In this section we collect some definitions and well known facts about del Pezzo surfaces, that we shall use in the sequel. For the proofs and many other results, we refer the reader to [Man86, Chapter 24 sq.].
A smooth projective surface defined over a field is del Pezzo when its anticanonical divisor is ample. Its degree is the self-intersection number .
We assume in the following. We know from [Man86, Theorems 24.4, 24.5] that such a surface is isomorphic (over the algebraic closure ) to the blow-up of the projective plane at points in general position, i.e. such that no three of them are collinear, and no six lie on a conic. In this case, the anticanonical divisor is very ample; the space of its global sections has dimension
and it defines an embedding of into , whose image has degree . The image of this embedding is called the anticanonical model of .
If is a birational morphism, and is a del Pezzo surface, then is a del Pezzo surface from [Man86, Theorem 24.5.2 (i)].
Let be as above; it is isomorphic (over ) to the blowup of the projective plane at . Let denote the pullback of the class of a line in and , denote respectively the class of the exceptional divisor of the blowup at . From [Man86, Theorems 25.1] (see also [Har77, Proposition V.3.2]), we know that the geometric Picard lattice is free of rank , with basis . The intersection pairing is defined by
and the class of the canonical divisor is
In the following, we assume that is defined over . Associated to is a representation of on that respects the intersection pairing and the canonical divisor. If is the image of under this representation, then we know from a result of Weil [Man86, Theorem 27.1] that the number of rational points of is given by
Moreover, since is projective and smooth and the ground field is finite, we have
, and the Picard rank is the multiplicity of the eigenvaluefor .
Following Dolgachev [Dol12, Section 8.2], we define the lattice as the orthogonal of the canonical divisor in . This is a root lattice, whose Weyl group is denoted . The image of the above Galois representation lies in , and it is a finite quotient of , thus a cyclic subgroup.
Following Manin [Man86], we define the type of the del Pezzo surface as the conjugacy class of the element in .
3.1. Isomorphism classes of del Pezzo surfaces of degrees five and six over a finite field
For a del Pezzo surface of degree or over a perfect field, the types of Manin are actually isomorphism classes (see [Sko01, Theorem 3.1.3] for degree , degree is treated in Section 4.3). When the field is finite, we deduce that the isomorphism classes of such del Pezzo surfaces correspond to the conjugacy classes of elements in the Weyl group .
3.1.1. Del Pezzo surfaces of degree
When , the root lattice is , and its Weyl group is
where denotes the symmetric group on letters and denotes the dihedral group of order , generated by the symmetries with respect to the following roots which form a principal system:
We have for and . Table 1 summarizes the different types of Del Pezzo surfaces with the corresponding Weyl conjugation classes.
|Type||Weyl classes||Eigenvalues of||Picard rank|
3.1.2. Del Pezzo surfaces of degree
When , the root lattice is , and its Weyl group is the symmetric group on letters, generated by the symmetries , , with respect to the roots
We identify these symmetries respectively to the transpositions and . The conjugacy classes in identify to the partitions of , and we get the classification summarised in Table 2.
|Type||Weyl classes||Eigenvalues of||Picard rank|
We see that there exists exactly one isomorphism class of del Pezzo surface having Picard rank in each degree or , corresponding respectively to the types and . We construct explicitely surfaces of these two types in the following sections.
We end this section with a technical result, that will be useful when we estimate the minimum distance of the codes. It is close to [LS18, Theorem 3.3].
Assume that is a del Pezzo surface over , with . If is an anticanonical curve that is not absolutely irreducible, then we have .
Let be the decomposition of into absolutely irreducible components. Since and generates the Picard group of , is irreducible over . We deduce that we must have , and the components are cyclically permuted by . As a consequence, every rational point must be at the intersection of all the ’s.
Assume that contains a rational point; from what we have just said, we must have for any . In the Picard group, we get , and the arithmetic genera satisfy [Har77, Ex V.1.3]
Since the ’s are absolutely irreducible, their arithmetic genera are nonnegative and hence, we get
If , then and contains at most two rational points. If , then for any , and contains at most one rational point. ∎
4. Anticanonical codes on some degree six del Pezzo surfaces
In this section, we construct some degree six del Pezzo surfaces with Picard rank one over any finite field, then we determine the parameters of the anticanonical codes on these surfaces.
4.1. Construction of the surface
two conjugate points in ,
three conjugate points in .
Recall that five points in the projective plane are in general position when no three of them are collinear. For any prime power it is possible to choose as above in general position: from [BFL16, Lemma 2.4] it is sufficient to choose them on a smooth conic; such a curve exists for any . Now if denotes a smooth conic in the projective plane, we have for any , and has points defined over but not over , and points defined over but not over .
Let denote the surface obtained from after blowing up the points , the corresponding exceptional divisors, and the composition of the five blowups. Since is stable under the action of , the map and the surface are defined over .
Applying the results in Section 3 to , we get the following properties. It is a degree del Pezzo surface with geometric Picard lattice , and canonical class . From our choice of the ’s, the map acts on the ’s by the permutation . As a consequence, the Picard lattice of is
and we get . Thus has points.
In the following, we denote by the line , and by the conic passing through in ; note that is unique since we assumed the points to be in general position, and both curves are defined over from our choice of the ’s. Let and denote the respective strict transforms of and in ; they are irreducible curves defined over . Their images in satisfy
We get , and they have arithmetic genus zero from the adjunction formula. Moreover they are disjoint since .
From Castelnuovo’s contractibility criterion [Bad01, Theorem 3.30], [Man86, Theorem 21.5], there exists a smooth projective surface , and a birational morphism contracting and to points and . In other words, the map is the composition of the blowups of at and , and , are the corresponding exceptional divisors. We have the diagram
The geometric Picard lattice identifies to the sublattice of generated by the classes
which satisfy , , and , .
We use [Har77, Proposition V.3.2]. Since is the composition of two blowups, the map from to is an isometry for the intersection pairing; as this pairing is non degenerate, it is an injection, and identifies to its image.
The classes of the exceptional divisors are and ; as a consequence, the image is the orthogonal of in . A class is in the orthogonal if, and only if
When we look for classes having self-intersection , we get the condition
We get the class , and any class orthogonal to must satisfy . The class satisfies the three equalities above, and orthogonality adds the equation . We get , and the last equation . The five equations above finally give . The classes clearly form a basis of from their intersection products. ∎
We determine the canonical divisor class of .
In the Picard lattice of , the class of the canonical divisor is
From [Har77, Proposition V.3.3], we have in . We get , and the result comes from the identification of the lemma above. ∎
We are ready to prove the main properties of our surface.
The surface has the following properties
it is a degree del Pezzo surface;
its Picard lattice has rank one, and is generated by ;
it is defined over , and has rational points.
As is a birational morphism from a del Pezzo surface, we know from Section 3 that is a del Pezzo surface. Its degree is .
Recall that we have . From the description of the action of on the ’s, we deduce that satisfies if, and only if we have and , i.e. . This proves assertion (ii).
Finally, is defined over since the canonical class is. To compute the number of its rational points, we use Weil’s result from Section 3. The matrix of the action of Frobenius on , with respect to the basis is
whose trace is . ∎
4.2. Anticanonical codes
Here we determine the parameters of the evaluation code .
Its length is from Proposition 4.3 (iii). The dimension of the space of global sections is from Section 3. We will see below that the evaluation map is injective for any , and in this case we get a code of dimension .
Let . Since generates the Picard lattice of , the curve is irreducible over . If it is absolutely irreducible, the adjunction formula gives
Thus, has arithmetic genus and contains at most rational points.
If is irreducible but not absolutely irreducible, we apply Lemma 3.2. Hence, the maximal number of points of a section comes from an absolutely irreducible one, and the minimum distance is at least . This number is positive as long as , and we get
Assume we have . Then the code has parameters
For , magma [BCP97] calculations give a minimum distance equal to , instead of , for all (random) choices of the points we made. Actually the anticanonical system does not carry any maximal elliptic curve, as we shall prove in Section 4.4. This is no longer true when : the minimum distances we observe are those given in the above Proposition.
4.3. Automorphisms of the surface
Asking magma for the automorphism groups of these codes, we get respectively groups of order for . Among these are the multiplications by scalars. The aim of this section is to show that the remaining automorphisms come from automorphisms of the surface.
We first describe the geometric group of automorphisms of the split degree del Pezzo surface obtained by blowing up the projective plane at the points , and . Note that is a toric variety whose maximal torus we note . From [Dol12, Theorem 8.4.2], we have
where is the dihedral group of order , which is the Weyl group of isometries of the geometric Picard lattice of a degree del Pezzo surface.
We first describe the action of these automorphisms on the maximal torus ; since the group is generated by the permutations (of the projective coordinates) , , and the standard quadratic transform , we get
and acts by .
For any and , set the composition of the above actions. One easily verifies the following assertions
we have , where is the image of by the action of described above.
the action of on is given by , since the surface is split and the action of the Weyl group is defined over .
We now determine , which gives the isomorphism classes of degree del Pezzo surfaces over (all these surfaces become isomorphic over the algebraic closure ). We have a short exact sequence (with a trivial Galois action on the diedral group)
which is split by the map . From [Ser94, I.5.5 Proposition 38], we have a map
and, by functoriality of , the splitting above ensures us that this map is surjective. In order to show that it is injective, it is sufficient to show that for any twist , the cohomology group vanishes [Ser94, I.5.5 Corollaire 2]. But this is a consequence of Lang’s theorem [Ser94, III.2.3 Théorème 1’] since in any case the twist is a smooth connected algebraic group.
Note that, since the groups and are not abelian, the ’s are not groups but only pointed sets. For this reason, the injectivity of the map cannot be proved using Hilbert 90 Theorem.
We deduce that the set corresponds to the set of conjugacy classes of elements of the group . With this at hand, we are ready to prove the following statement.
There are automorphisms of the surface defined over .
The eigenvalues of the matrix of (4.1) are . Therefore, this matrix has order and, from our calculation of , the degree del Pezzo surface constructed above corresponds to an element of order in . There is only one up to conjugacy, and we choose . In other words, if is an isomorphism over , the cocyle corresponding to , sends on .
Let denote an automorphism of over ; from it we construct an automorphism of . Now is defined over if and only if we have , that is
If we write as above, we get the condition
and the automorphisms of are the , where is in the centralizer of , which is the order subgroup of generated by , and satisfies , , i.e. , . ∎
4.4. Improving the minimum distance over the field with four elements
Assume that the anticanonical linear system carries a maximal elliptic curve defined over , i.e. with . This curve must be smooth since its geometric genus equals its arithmetic genus.
We begin with a lemma about the automorphisms of the surface and their action on the curve .
The automorphism group of satisfies the following properties.
The group contains an element of order , which permutes cyclically the set .
There exists an such that contains the points and .
The order of is from Proposition 4.7; thus this group contains an element of order from a theorem of Cauchy. Since , either permutes cyclically the rational points of , or its fixes all of them.
The automorphism preserves the exceptional divisors of ; as a consequence, it induces an automorphism on the complementary of these divisors in . As we have seen in 4.3, the surface is isomorphic to , and the image of under such an isomorphism is the maximal torus .
Thus induces an automorphism of , i.e. an element in ; since has order , its image must lie in and have the form . Since the automorphism has no fixed point on the maximal torus, does not have any fixed point on .
Now the exceptional divisors of are the images under of the strict transforms of the lines , , . They are cyclically permuted by the action of the Galois group ; any rational point on one of these divisors must lie on all, which does not happen. We get , and has no fixed point in this set. This proves the first assertion.
From above, since and lie in , there exists some such that ; replacing by we assume that in the following. If we write for some , then at least two of the ’s are consecutive, say ; the automorphism satisfies the requirements of the second assertion. ∎
Thus we can assume that is a maximal curve containing and . Denote by its strict transform under ; since is smooth, it has multiplicity one at and , and we have in . Moreover induces an isomorphism between the curves and .
The curve lies in the anticanonical system of ; it is smooth and since for any , we have , then is transversal to the exceptional divisors . As a consequence, it is isomorphic to its image under , which is a smooth cubic passing through the points .
Thus the curve is a smooth elliptic curve in having rational points. Its Frobenius eigenvalues must be equal to , and we have
But contains and which are defined over but not over , a contradiction. We deduce that the anticanonical linear system does not carry any maximal curve over , and the minimum distance of the anticanonical code is at least . Consequently, thanks to the Griesmer bound, we get the following result.
The code over has parameters .
5. Anticanonical codes on some degree five del Pezzo surfaces
In this section, we construct some degree five del Pezzo surfaces with Picard rank one over any finite field, then we determine the parameters of the anticanonical codes on these surfaces. Many arguments are similar to those of the preceding section, for this reason we shall skip some proofs.
A new feature here is that we try to be as constructive as possible: we describe explicit constructions of the codes and their automorphisms.
5.1. Construction of the surface
We denote by the projective plane defined over , and we choose
five conjugate points in . We assume that they are in general position, i.e. no three are collinear. This is possible for any since a smooth conic in has points defined over but not over .
Let be the surface obtained by blowing up the plane at the ’s. Once again we get a degree del Pezzo surface and denote by the pullback of the class of a line in and by the exceptional divisors. The descriptions of the geometric Picard lattice of , and its intersection pairing are the same as in Section 4, as for its canonical divisor. The difference here is that the map acts on the ’s as the permutation ; as a consequence, the Picard lattice has rank , and is generated by and .
Let denote the unique conic passing through ; it is defined over . Its strict transform in is an irreducible curve, whose class satisfies
Once again this curve has self-intersection and arithmetic genus zero.
Applying Castelnuovo’s contractibility criterion, we obtain a smooth surface by contracting the curve in , and a birational morphism . If we set , then is the blowup of at , with exceptional divisor .
The geometric Picard lattice can be identified to the orthogonal in of the class of (see Lemma 4.1). After similar calculations, we get the following “orthonormal” basis for :
These classes satisfy , for any and for any .
Note that the classes contain respectively the strict transforms of the lines and the corresponding curves satisfy Castelnuovo’s contractibility criterion.
The canonical divisor of satisfies , and we get via the above identification. The matrix of the image of Frobenius acting on with respect to the basis is