On space maximal curves

. Any maximal curve X is equipped with an intrinsic embedding π : X → P r which reveal outstanding properties of the curve. By dealing with the contact divisors of the curve π ( X ) and tangent lines, in this paper we investigate the ﬁrst positive element that the Weierstrass semigroup at rational points can have whenever r = 3 and π ( X ) is contained in a cubic surface.


Introduction
Throughout this paper, F stands for the finite field F q 2 of order q 2 . A projective, geometrically irreducible, non-singular algebraic curve X defined over F of genus g = g(X ) is said to be F-maximal if the number of its F-rational points attains the Hasse-Weil upper bound; that is, #X (F) = q 2 + 1 + 2q · g .
Apart from being interesting mathematical objects by their own, these curves have been extensively studied as they are of great interest in Coding Theory, Cryptography and related areas; see for example the books [24], [16], [14].
Let X be an F-maximal curve of genus g. Then the numerator of the Zeta function of X is the polynomial L(t) = (1 + qt) 2g and hence h(t) = t 2g L(t −1 ) = (t + q) 2g is the characteristic polynomial of certain endomorphismΦ on the Jacobian J of X . This map is uniquely determined by the F-Frobenius morphism Φ : X → X in such a way that ι • Φ =Φ • ι, where ι : X → J is the natural embedding given by P → [P − P 0 ] with P 0 ∈ X (F). It turns out that Φ is semisimple and so the following linear equivalence (sometimes called the fundamental equivalence) on X arises (see [16,Thm. 10.1,Thm. 9.79]): (q + 1)P 0 ∼ qP + Φ(P ) , P ∈ X . (1) This suggests the study of the (complete) linear series D X := |(q + 1)P 0 | (sometimes called the Frobenius linear series of X ) whose definition clearly does not depend on the choice of the F-rational point P 0 . As a matter of fact, several arithmetical and geometrical properties of maximal curves are revealed through this linear series (loc. cit.). In particular, D X is very ample [7,Prop. 1.9], [18,Thm. 2.5] which means that the morphism associated to D X π D X : X → P r is an embedding, where r = r(X ) ≥ 2 is the projective dimension of D X (sometimes called the Frobenius dimension of X ), and P r is the projective r-space over the algebraic closure of F.
can be used to explain partially Proposition 1.1 as it gives the first general constrain between g(X ) and the pair (q, r) (see [16,Cor. 10.25]): Notice that the function F (q, r) satisfies F (q, r) ≤ F (q, s) for r ≥ s; in particular, g(X ) ≤ F (q, 3) = (q − 1) 2 /4 provided that r(X ) ≥ 3. Then, with g 1 := (q −1) 2 /4 , by Proposition 1.1 the spectrum for the genera of F-maximal curves, namely the set We recall that g 0 is the well-known Ihara's bound on the genus of F-maximal curves [15]. One of the main problems in Curve Theory Over Finite Fields is the computation of M(q 2 ); in general one cannot expect to give a full answer to this matter but improvements on (4) can be expected as far as improvements on Castelnuovo's genus bound of curves in P r are known.
In view of Proposition 1.1 it is natural to investigate space F-maximal curves with respect to D X ; that is, those with r(X ) = 3. Here a natural way of bounding g(X ), which generalizes Castelnuovo's method, is by looking at the degree d ≥ 2 of surfaces S ⊆ P 3 such that π(X ) ⊆ S where π = π X is as in (2); cf. [12], [21]. We have the following Halphen-Ballico result (see [3]) which deals with the case of quadrics. Let g 1 be as in (4) and set g 2 := (q 2 − q + 4)/6 ; then π(X ) is contained in a quadric in P 3 provided that g 2 < g(X ) ≤ g 1 .
Now the F-maximal property of X implies certain constrains on the first positive element m 1 (P ) of the Weierstrass semigroup H(P ) at some P ∈ X (F), and (4) admits the folloing improvement [18]: An analogue of Proposition 1.1 emerges, namely , [1], [17], [18]) Let X be an F-maximal curve. The following sentences are equivalent: (1) X is isomorphic to a quotient of H by certain involution; (2) g(X ) = g 1 ; (3) π(X ) is contained in a quadric; (4) There exists P ∈ X (F) such that the first positive element of H(P ), the Weierstrass semigroup at P , equals (q + 1)/2 .
The starting points of our result are in fact Propositions 1.1, 1.2 above. Under condition (7) below, the main result in this paper is Corollary 2.6, where a hypothesis on a cubic surface is considered; in this way a weak version of the aforementioned propositions is obtained. We always assume q > 7; cf. [2].
Conventions. P s stands for the projective s-space over the algebraic closure of the base field. For a point P in a curve, H(P ) denotes the Weierstrass semigroup at P ; m 1 (P ) is the first positive element of H(P ).

Maximal curves and cubic surfaces
Let X be an F-maximal curve, P 0 ∈ X (F) and D = D X = |(q + 1)P 0 | the liner series introduced in Section 1; i.e., it is the set of effective divisors on X which are linearly equivalent to the divisor (q + 1)P 0 . We always assume g(X ) > 0; taking into consideration (6) and Propositions 1.1, 1.2 above, we also assume: Remark 2.1. Let X be an F-maximal curve. From (3) and Proposition 1.1, a sufficient condition to have Let π = π D : X → P 3 be the morphism associated to D.
Thus for each i = 0, 1, 2, 3 there are rational functions on X , h i : therefore at P ∈ X (F), j 3 (P ) = q + 1 and the first positive element m 1 (P ) of H(P ) and j 2 (P ) are related to each other by the equation Remark 2.3. For the linear system D above and any P ∈ X , the (D, P ) orders can be ordered as a sequence j 0 (P ) < j 1 (P ) < j 2 (P ) < j 3 (P ) ≤ q + 1 with j 0 (P ) = 0 as D is base-point-free. Relation (1) shows that 1 and q are (D, P )-orders for P ∈ X (F). Thus for such points j 1 (P ) = 1 and j 3 (P ) = q (as g(X ) > 0).
Lemma 2.4. Let X be an F-maximal curve satisfying (7) and let P ∈ X (F).
Theorem 2.5. Let X be an F-maximal curve satisfying (7). Suppose that π(X ) is contained in a cubic surface S.
Now we can state the main result in this paper.
Corollary 2.6. Let X be an F-maximal curve as in Theorem 2.5. Then the multiplicity m 1 (P ) of the Weierstrass semigroup H(P ) at P ∈ X (F) do satisfy In addition, if q is even and g(X ) > q 2 /8, then m 1 (P ) = (q + 2)/2.
Remark 2.7. Notation as in Remark 2.3. For the linear series D, a basic result is that for almost P ∈ X , the sequence j 0 (P < j 1 (P ) < j 2 (P ) < j 3 (P ) is constant (so called order sequence of D) cf. [25, p. 5]). In Remark 2.3 we noticed that j 0 (P ) = 0, j 1 (P ) = 1, j 3 (P ) = q for P ∈ X (F) and thus the order sequence of D is of type 0 < 1 < 2 < q.
Remark 2.8. Let X be an F-maximal curve such that (7) holds; in particular, we identify X with a non-degenerate projective curve in P 3 and we can apply the aforementioned Castelnuovo and Halphen-Ballico results as they are true in positive characteristic [3]. We look forward a result of type: There exists a polynomial (of one indeterminate) Let q be large, says q ≥ 107. If then there exists a surface S of degree 2 or 3 such that C ⊆ S.
Question 2.10. Is Remark 2.9 true in positive characteristic?

Examples
In this section we illustrate Corollary 2.6. Notation as above; in particular, H is the Hermitian curve over F = F q 2 defined by v q+1 = u q+1 + 1. Let π : H → P 2 be a non-trivial morphism over F and X the non-singular model of the plane curve π(H); then π can be lifted to a morphism H → X , which we still denote by π. In this case, the curve X is also F-maximal (see e.g. [19]).
Next we shall compute the Weierstrass semigroup H(P ) at certain points of X ; we start by computing some principal divisors on X via tools from [24].
Moreover by Remark 2.7 the order sequence of X is 0 < 1 < 2 < q and thus there is also a point P ∈ X (F) with m 1 (P ) = q − 1 (see [4,Lemma 3.7]).
Remark 3.2. We can construct explicit and outstanding AG one-point codes based on the curve in Example 3.1 by taking into consideration the telescopic property of H(R a ); cf. [13,Sect. 5], [26,Sect. 5].
Let π : X → P 3 be the morphism associated to D. We are led to the following questions.   (4), for an F-maximal curve X we have that g(X ) = g 1 if and only if π(X ) is contained in a quadric and there isP ∈ X (F) with j 2 (P ) > 2.
We further assume the following properties: (a) π(X ) is contained in a cubic surface; (b) π : H → X is Galois of degree three.
Example 3.7. Here we present an F-maximal curve X with r(X ) = 3 such that π(X ) cannot be contained in a cubic surface, where π is the morphism associated to D. Indeed, we consider the so-called GK-curve [11] whose Weierstrass semigroups at rational points were computed in [6]. This curve is defined over F = F q 2 with q = 3 . For > 2 this is the first example of an F-maximal curve that cannot be dominated by H (loc. cit.) On this curve there isP ∈ X (F) such that m 1 (P ) = 3 − 2 + [11, Sect. 4], and therefore, according to Corollary 2.6, π(X ) cannot be contained in a cubic. We notice that the genus of X is g(X ) = 1 2 ( 5 − 2 3 + 2 )/2 and so it does not satisfies Remark 2.9. Further examples can be found in [26].
We end this paper with the following: Question 3.8. Let X be an F-maximal curve with r(X ) = 3. Suppose that π(X ) ⊆ S, where S is a surface of degree d ≥ 2. Let P ∈ X (F) and suppose g(X ) large enough. Then m 1 (P ) = (q + 1) − q+i d or m 1 (P ) = q − j for some i = 1, . . . , d, j = 2, . . . , d. Are all these cases possible?