For every $q=n^3$ with $n$ a prime power greater than $2$, the GK-curve is an $\mathbb F_{q^2}$-maximal curve that is not $\mathbb F_{q^2}$-covered by the Hermitian curve. In this paper some Galois subcovers of the GK curve are investigated. We describe explicit equations for some families of quotients of the GK-curve. New values in the spectrum of genera of $\mathbb F_{q^2}$-maximal curves are obtained. Finally, infinitely many further examples of maximal curves that cannot be Galois covered by the Hermitian curve are provided.
We determine all genus 2 curves, defined over $\mathbb C$, which have simultaneously degree 2 and 3 elliptic subcovers. The locus of such curves has three irreducible 1-dimensional genus zero components in $\mathcal M_2$. For each component we find a rational parametrization and construct the equation of the corresponding genus 2 curve and its elliptic subcovers in terms of the parameterization. Such families of genus 2 curves are determined for the first time. Furthermore, we prove that there are only finitely many genus 2 curves (up to $\mathbb C$-isomorphism) defined over $\mathbb Q$, which…
In this work, subcovers $\mathcal{X}_{n,r}^s$ of the curve $\mathcal{X}_{n,r}$ are constructed, the Weierstrass semigroup $H(P_\infty)$ at the point $P_\infty \in \mathcal{X}_{n,r}^s$ is determined and the corresponding one-point AG codes are investigated. Codes establishing new records on the parameters with respect to the previously known ones are discovered, and $108$ improvements on MinT tables are obtained.
We show that the generalized Giulietti-Korchmáros curve is not a Galois subcover of the Hermitian curve over $\mathbb{F}_{q^{2n}}$. This answers a question raised by Garcia, Güneri and Stichtenoth.
In 2017 Skabelund constructed two new examples of maximal curves $\tilde{\mathcal{S}}_q$ and $\tilde{\mathcal{R}}_q$ as covers of the Suzuki and Ree curves, respectively. The resulting Skabelund curves are analogous to the Giulietti-Korchmáros cover of the Hermitian curve. In this paper a complete characterization of all Galois subcovers of the Skabelund curves $\tilde{\mathcal{S}}_q$ and $\tilde{\mathcal{R}}_q$ is given. Calculating the genera of the corresponding curves, we find new additions to the list of known genera of maximal curves over finite fields.
In this article we construct for any prime power $q$ and odd $n \ge 5$, a new $\mathbb{F}_{q^{2n}}$-maximal curve $\mathcal X_n$. Like the Garcia--G\" uneri--Stichtenoth maximal curves, our curves generalize the Giulietti--Korchmáros maximal curve, though in a different way. We compute the full automorphism group of $\mathcal X_n$, yielding that it has precisely $q(q^2-1)(q^n+1)$ automorphisms. Further, we show that unless $q=2$, the curve $\mathcal{X}_n$ is not a Galois subcover of the Hermitian curve. Finally, we find new values of the genus spectrum of $\mathbb{F}_{q^{2n}}$-maximal curves, …
Genus two curves covering elliptic curves have been the object of study of many articles. For a fixed degree $n$ the subloci of the moduli space $\mathcal M_2$ of curves having a degree $n$ elliptic subcover has been computed for $n=3, 5$ and discussed in detail for $n$ odd; see \cite{Sh1, SV2, Fr, FK}. When the degree of the cover is even the case in general has been treated in \cite{PRS}. In this paper we compute the sublocus of $\mathcal M_2$ of curves having a degree 4 elliptic subcover.
A genus 2 curve $C$ has an elliptic subcover if there exists a degree $n$ maximal covering $ψ: C \to E$ to an elliptic curve $E$. Degree $n$ elliptic subcovers occur in pairs $(E, E')$. The Jacobian $J_C$ of $C$ is isogenous of degree $n^2$ to the product $E \times E'$. We say that $J_C$ is $(n, n)$-split. The locus of $C$, denoted by $Ł_n$, is an algebraic subvariety of the moduli space $\M_2$. The space $Ł_2$ was studied in Shaska/Völklein and Gaudry/Schost. The space $Ł_3$ was studied in Shaska (2004) were an algebraic description was given as sublocus of $\M_2$. In this survey we give a br…
The Deligne-Lusztig curves associated to the algebraic groups of type $^2A_2$, $^2B_2$, and $^2G_2$ are classical examples of maximal curves over finite fields. The Hermitian curve $\mathcal H_q$ is maximal over $\mathbb F_{q^2}$, for any prime power $q$, the Suzuki curve $\mathcal S_q$ is maximal over $\mathbb F_{q^4}$, for $q=2^{2h+1}$, $h\geq1$ and the Ree curve $\mathcal R_q$ is maximal over $\mathbb F_{q^6}$, for $q=3^{2h+1}$, $h\geq0$. In this paper we show that $\mathcal S_8$ is not Galois covered by $\mathcal H_{64}$. We also give a proof for an unpublished result due to Rains and Ziev…
In this paper we compute the automorphism group of the curves $\mathcal{X}_{a,b,n,s}$ and $\mathcal{Y}_{n,s}$ introduced in Tafazolian et al. in 2016 as new examples of maximal curves which cannot be covered by the Hermitian curve. They arise as subcovers of the first generalized GK curve (GGS curve). As a result, a new characterization of the GK curve, as a member of this family, is obtained.
A criterion for the existence of a birational embedding with two Galois points for quotient curves is presented. We apply our criterion to several curves, for example, some cyclic subcovers of the Giulietti-Korchmaros curve or of the curves constructed by Skabelund. They are new examples of plane curves with two Galois points.
We determine the full automorphism group of two recently constructed families $\tilde{\mathcal{S}}_q$ and $\tilde{\mathcal{R}}_q$ of maximal curves over finite fields. These curves are covers of the Suzuki and Ree curves, and are analogous to the Giulietti-Korchmáros cover of the Hermitian curve. We also show that $\tilde{\mathcal{S}}_q$ is not Galois covered by the Hermitian curve maximal over $\mathbb{F}_{q^4}$, and $\tilde{\mathcal{R}}_q$ is not Galois covered by the Hermitian curve maximal over $\mathbb{F}_{q^6}$. Finally, we compute the genera of many Galois subcovers of $\tilde{\mathcal{…
In this paper we characterize the genera of those quotient curves $\mathcal{H}_q/G$ of the $\mathbb{F}_{q^2}$-maximal Hermitian curve $\mathcal{H}_q$ for which $G$ is contained in the maximal subgroup $\mathcal{M}_1$ of ${\rm Aut}(\mathcal{H}_q)$ fixing a self-polar triangle, or $q$ is even and $G$ is contained in the maximal subgroup $\mathcal{M}_2$ of ${\rm Aut}(\mathcal{H}_q)$ fixing a pole-polar pair $(P,\ell)$ with respect to the unitary polarity associated to $\mathcal{H}_q(\mathbb{F}_{q^2})$. In this way several new values for the genus of a maximal curve over a finite field are obtaine…
In this paper we solve three open problems on maximal curves with Frobenius dimension 3. In particular, we prove the existence of a maximal curve with order sequence (0,1,3,q).
We discuss sufficient conditions for a given curve to be covered by a maximal curve with the covering being unramified; it turns out that the given curve itself will be also maximal. We relate our main result to the question of whether or not a maximal curve is covered by the Hermitian curve. We also provide examples illustrating the results.
We construct new covers of the Suzuki and Ree curves which are maximal with respect to the Hasse-Weil bound over suitable finite fields. These covers are analogues of the Giulietti-Korchmáros curve, which covers the Hermitian curve and is maximal over a base field extension. We show that the maximality of these curves implies that of certain ray class field extensions of each of the Deligne-Lusztig curves. Moreover, we show that the Giulietti-Korchmáros curve is equal to the above-mentioned ray class field extension of the Hermitian curve.
In this paper, we compute the fundamental group of the complement of linear torus curves of maximal contact and we show that it is isomorphic that of generic linear torus curves.
Some linear codes associated to maximal algebraic curves via Feng-Rao construction are investigated. In several case, these codes have better minimum distance with respect to the previously known linear codes with same length and dimension.
In this note, we give an overview of a new technique for studying Brill--Noether curves in projective space via degeneration. In particular, we give a roadmap to the proof of the Maximal Rank Conjecture.
The genus of a maximal curve over a finite field with r^2 elements is either g_0=r(r-1)/2 or less than or equal to g_1=(r-1)^2/4. Maximal curves with genus g_0 or g_1 have been characterized up to isomorphism. A natural genus to be studied is g_2=(r-1)(r-3)/8, and for this genus there are two non-isomorphism maximal curves known when r \equiv 3 (mod 4). Here, a maximal curve with genus g_2 and a non-singular plane model is characterized as a Fermat curve of degree (r+1)/2.
In this work we present explicit examples of maximal and minimal curves over finite fields in odd characteristic. The curves are of Artin-Schreier type and the construction is closely related to quadratic forms from $\mathbb{F}_{q^n}$ to $\mathbb{F}_q$.
The maximal number of automorphisms of a Mumford curve over a non-archimedean valued field of positive characteristic is known. In this note, the unique family of curves that attains this bound in genus not equal to 5,6,7 or 8 is explicitly determined, as is its automorphism group.
The purpose of this paper is to extend the symmetry of maximals of the ring of a germ of reducible plane curve proved by Delgado to a relation between the relative maximals of a fractional ideal and the absolute maximals of its dual for any admissible ring. In particular, it includes the case of germs of reduced reducible curve of any codimension. We then apply this symmetry to characterize the elements in the set of values of a fractional ideal from some of its projections and the irreducible absolute maximals of the dual ideal.
The classification of maximal plane curves of degree $3$ over $\mathbb{F}_4$ will be given, which complements Hirschfeld-Storme-Thas-Voloch's theorem on a characterization of Hermitian curves in $\mathbb{P}^2$. This complementary part should be understood as the classification of Sziklai's example of maximal plane curves of degree $q-1$ over $\mathbb{F}_q$. Although two maximal plane curves of degree $3$ over $\mathbb{F}_4$ up to projective equivalence over $\mathbb{F}_4$ appear, they are birationally equivalent over $\mathbb{F}_4$ each other.
In this paper, we study a Howe curve $C$ in positive characteristic $p \geq 3$ which is of genus 3 and is hyperelliptic. We will show that if $C$ is superspecial, then its standard form is maximal or minimal over $\mathbb{F}_{p^2}$ without taking its $\mathbb{F}_{p^2}$-form.
We study families of n-gonal curves with maximal variation of moduli, which have a rational section. Certain numerical results on the degree of the modular map are obtained for such families of hyperelliptic and trigonal curves. In the last case we use the description of the relative Picard group of the universal family of trigonal curves.
The genus g of an F_{q^2}-maximal curve satisfies g=g_1:=q(q-1)/2 or g\le g_2:= [(q-1)^2/4]. Previously, such curves with g=g_1 or g=g_2, q odd, have been characterized up to isomorphism. Here it is shown that an F_{q^2}-maximal curve with genus g_2, q even, is F_{q^2}-isomorphic to the nonsingular model of the plane curve \sum_{i=1}^{t}y^{q/2^i}=x^{q+1}, q=2^t, provided that q/2 is a Weierstrass non-gap at some point of the curve.
In 2014 A. Degtyarev, I. Itenberg and the author gave a description, up to fiberwise equivariant deformations, of maximally inflected real trigonal curves of type~I (over a base $ B $ of an arbitrary genus) in terms of the combinatorics of sufficiently simple graphs and for $ B=\mathbb{P}^1 $ obtained a complete classification of such curves. In this paper, the mentioned results are extended to maximally inflected real trigonal curves of type~II over $ B=\mathbb{P}^1 $. As usual, the results for such curves are extended to real Jacobian elliptic surfaces with all singular fibers real.
In this paper we determine the generalized Weierstrass semigroup $ \widehat{H}(P_{\infty}, P_1, \ldots , P_{m})$, and consequently the Weierstrass semigroup $H(P_{\infty}, P_1, \ldots , P_{m})$, at $m+1$ points on the curves $\mathcal{X}_{a,b,n,s}$ and $\mathcal{Y}_{n,s}$. These curves has been introduced in Tafazolian et al. as new examples of curves which cannot be covered by the Hermitian curve.
Maximal curves with respect to quadratic extensions over finite fields
Institut de Mathématiques de Toulon - IMATH, Université de Toulon（土伦大学数学研究所）；Institut de Mathématiques de Marseille - I2M, Aix Marseille Univ, UMR 7373 CNRS（马赛大学数学研究所）；INSPE Nice-Toulon, Université Côte d’Azur（尼斯-土伦高等师范学院，蔚蓝海岸大学）
Some new results on plane F_{q^2}-maximal curves are stated and proved. It is known that the degree d of such curves is upper bounded by q+1 and that d=q+1 if and only if the curve is F_{q^2}-isomorphic to the Hermitian. We show that d\le q+1 can be improved to d\le (q+2)/2 apart from the case d=q+1 or q\le 5. This upper bound turns out to be sharp for q odd. We also study the maximality of Hurwitz curves of degree n+1. We show that they are F_{q^2}-maximal if and only if (q+1) divides (n^2-n+1). Such a criterion is extended to a wider family of curves.
We prove the following statement, which has been conjectured since 1985: There exists a constant $K$ such that for all natural numbers $d,g$ with $g\le Kd^{3/2}$ there exists an irreducible component of the Hilbert scheme of $\mathbb{P}^3$ whose general element is a smooth, connected curve of degree $d$ and genus $g$ of maximal rank.
Let $\mathbb{F}$ be the finite field of order $q^2$, $q=p^h$ with $p$ prime. It is commonly atribute to J.P. Serre the fact that any curve $\mathbb{F}$-covered by the Hermitian curve $\mathcal{H}_{q+1}:\, y^{q+1}=x^q+x$ is also $\mathbb{F}$-maximal. Nevertheless, the converse is not true as the Giulietti-Korchmáros example shows provided that $q>8$ and $h\equiv 0\pmod{3}$. In this paper, we show that if an $\mathbb{F}$-maximal curve $\mathcal{X}$ of genus $g\geq 2$ where $q=p$ is such that $|Aut(\mathcal{X})|>84(g-1)$ then $\mathcal{X}$ is Galois-covered by $\mathcal{H}_{p+1}$. Also, we show t…
Let $\mathbb{F}_{q^2}$ be the finite field with $q^2$ elements. Most of the known $\mathbb{F}_{q^2}$-maximal curves arise as quotient curves of the $\mathbb{F}_{q^2}$-maximal Hermitian curve $\mathcal{H}_{q}$. After a seminal paper by Garcia, Stichtenoth and Xing, many papers have provided genera of quotients of $\mathcal{H}_q$, but their complete determination is a challenging open problem. In this paper we determine completely the spectrum of genera of quotients of $\mathcal{H}_q$ for any $q\equiv1\pmod4$.
In this paper, we study a Ciani curve $C: x^4 + y^4 + z^4 + rx^2y^2 + sy^2z^2 + tz^2x^2 = 0$ in positive characteristic $p \geq 3$. We will show that if $C$ is superspecial, then its standard form is maximal or minimal over $\mathbb{F}_{p^2}$ without taking its $\mathbb{F}_{p^2}$-form.
Previous results on genera g of F_{q^2}-maximal curves are improved: (1) Either g\leq (q^2-q+4)/6, or g=\lfloor(q-1)^2/4\rfloor, or g=q(q-1)/2; (2) The hypothesis on the existence of a particular Weierstrass point in \cite{at} is proved; (3) For q\equiv 1\pmod{3}, q\ge 13, no F_{q^2}-maximal curve of genus (q-1)(q-2)/3 exists; (4) For q\equiv 2\pmod{3}, q\ge 11, the non-singular F_{q^2}-model of the plane curve of equation y^q+y=x^{(q+1)/3} is the unique F_{q^2}-maximal curve of genus g=(q-1)(q-2)/6; (5) Assume \dim(\cD_\cX)=5, and char(\fq)\geq 5. For q\equiv 1\pmod{4}, q\geq 17, the Fermat c…
The ordinary algebraic curves of maximal rank are also the arithmetically Cohen-Maccaulay curves of minimal rank. We give sufficient conditions for such curves to exist as well as examples, generalizing results of [GHL] in the dimension three.
Fix positive integers d;m such that $(m^2+4m+6)/6 \leq d < (m^2+4m+6)/3$ (the so-called Range A for space curves). Let G(d;m) be the maximal genus of a smooth and connected curve, of degree d, $C \subset P^3$ such that $h^0(I_C(m-1)) = 0$. Here we prove that $G(d,m) = 1+(m-1)d -\binom{m+2}{3}$ if $m\ge 13.8\cdot 10^5$.
We study hyperelliptic curves arising from Chebyshev polynomials. The aim of this paper is to characterize the pairs $(q,d)$ such that the hyperelliptic curve $\cC$ over a finite field $\FF_{q^2}$ corresponding to the equation $y^2 = φ_{d}(x)$ is maximal over the finite field $\FF_{q^2}$ of cardinality $q^2$. Here $φ_{d}(x)$ denotes the Chebyshev polynomial of degree $d$. The same question is studied for the curves corresponding to $y^2=(x\pm 2) φ_{d}(x)$, and also for $y^2=(x^2-4)φ_d(x)$.
Let $C$ be a smooth projective curve and $W$ a symplectic bundle over $C$. Let $LQ_e (W)$ be the Lagrangian Quot scheme parametrizing Lagrangian subsheaves $E \subset W$ of degree $e$. We give a closed formula for intersection numbers on $LQ_e (W)$. As a special case, for $g \ge 2$, we compute the number of Lagrangian subbundles of maximal degree of a general stable symplectic bundle, when this is finite. This is a symplectic analogue of Holla's enumeration of maximal subbundles in [13].
We classify, up to isomorphism, maximal curves covered by the Hermitian curve \mathcal H by a prime degree Galois covering. We also compute the genus of maximal curves obtained by the quotient of \mathcal H by several automorphisms groups. Finally we discuss the value for the third largest genus that a maximal curve can have.
Let be a general curve of genus g embedded via a general linear series of degree d in P^r. The well-known Maximal Rank Conjecture asserts that the restriction maps H^0(O_{P^r}(m)) \to H^0(O_C(m) are of maximal rank; if known, this conjecture would determine the Hilbert function of C. In this paper, we prove an analogous statement for the hyperplane sections of unions general curves. More specifically, if H is a general hyperplane, we show that H^0(O_H(m)) \to H^0(O_{(C_1 \cup C_2 \cup \cdots \cup C_n) \cap H}(m)) is of maximal rank, except for some counterexamples when m = 2. As explained in a…
We prove that if $X$ is a complex projective K3 surface and $g>0$, then there exist infinitely many families of curves of geometric genus $g$ on $X$ with maximal, i.e., $g$-dimensional, variation in moduli. In particular every K3 surface contains a curve of geometric genus 1 which moves in a non-isotrivial family. This implies a conjecture of Huybrechts on constant cycle curves and gives an algebro-geometric proof of a theorem of Kobayashi that a K3 surface has no global symmetric differential forms.
We prove that curves in a non-primitive, base point free, ample linear system on a K3 surface have maximal variation. The result is deduced from general restriction theorems applied to the tangent bundle. We also show how to use specialisation to spectral curves to deduce information about the variation of curves contained in a K3 surface more directly. The situation for primitive linear systems is not clear at the moment. However, the maximal variation holds in genus two and can, in many cases, be deduced from a recent result of van Geemen and Voisin confirming a conjecture due to Matsushita.
A Mumford curve of genus g=5,6,7 or 8 over a non-archimedean field of characteristic p (such that if p=0, the residue field characteristic exceeds 5) has at most 12(g-1) automorphisms. In this paper, all curves that attain this bound and their automorphism groups (called of Lame type) are explicitly determined.
For each prime power $\ell$ the plane curve $\mathcal X_\ell$ with equation $Y^{\ell^2-\ell+1}=X^{\ell^2}-X$ is maximal over $\mathbb{F}_{\ell^6}$. Garcia and Stichtenoth in 2006 proved that $\mathcal X_3$ is not Galois covered by the Hermitian curve and raised the same question for $\mathcal X_\ell$ with $\ell>3$; in this paper we show that $\mathcal X_\ell$ is not Galois covered by the Hermitian curve for any $\ell>3$. Analogously, Duursma and Mak proved that the generalized GK curve $\mathcal C_{\ell^n}$ over $\mathbb{F}_{\ell^{2n}}$ is not a quotient of the Hermitian curve for $\ell>2$ and…
We show that a F_{q^2}-maximal curve of genus q(q-3)/6 in characteristic three is either a non-reflexive space curve of degree q+1, or it is uniquely determined up to F_{q^2}-isomorphism by a plane model of Artin-Schreier type
In 2017, D. Skabelund constructed a maximal curve over $\mathbb{F}_{q^4}$ as a cyclic cover of the Suzuki curve. In this paper we explicitly determine the structure of the Weierstrass semigroup at any point $P$ of the Skabelund curve. We show that its Weierstrass points are precisely the $\mathbb{F}_{q^4}$-rational points. Also we show that among the Weierstrass points, two types of Weierstrass semigroup occur: one for the $\mathbb{F}_q$-rational points, one for the remaining $\mathbb{F}_{q^4}$-rational points. For each of these two types its Apéry set is computed as well as a set of generator…
The second generalized GK maximal curves $\mathcal{GK}_{2,n}$ are maximal curves over finite fields with $q^{2n}$ elements, where $q$ is a prime power and $n \geq 3$ an odd integer, constructed by Beelen and Montanucci. In this paper we determine the structure of the Weierstrass semigroup $H(P)$ where $P$ is an arbitrary $\mathbb{F}_{q^2}$-rational point of $\mathcal{GK}_{2,n}$. We show that these points are Weierstrass points and the Frobenius dimension of $\mathcal{GK}_{2,n}$ is computed. A new proof of the fact that the first and the second generalized GK curves are not isomorphic for any $…
A well known result of Miyaoka asserts that a complex projective manifold is uniruled if its cotangent bundle restricted to a general complete intersection curve is not nef. Using the Harder-Narasimhan filtration of the tangent bundle, it can moreover be shown that the choice of such a curve gives rise to a rationally connected foliation of the manifold. In this note we show that, conversely, a movable curve can be found so that the maximal rationally connected fibration of the manifold may be recovered as a term of the associated Harder-Narasimhan filtration of the tangent bundle.
We construct some complex surfaces of general type with maximal Picard number. These examples arise as fibrations of genus two curves over quaternionic Shimura curves.
