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 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).
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 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 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, …
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 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.
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.
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 $…
Vector bundles in positive characteristics have a tendency to be destabilized after pulling back by the Frobenius morphism. In this paper, we closely examine vector bundles over curves that are, in an appropriate sense, maximally destabilized by the Frobenius morphism. Then we prove that such bundles of rank 2 exist over any curve in characteristic 3, and are unique up to twisting by a line bundle. We also give an application of such bundles to the study of ample vector bundles, which is valid in all characteristics.
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.
We give a simplified and more algebraic proof of the finiteness of the families of Calabi-Yau n-folds with non-vanishing of Yukawa-coupling over a fixed base curve and with fixed degeneration locus. We also give a generalization of this result. Our method is variation of Hodge structure and poly-stability of Higgs bundles.
Using an Euclidean approach, we prove a new upper bound for the number of closed points of degree 2 on a smooth absolutely irreducible projective algebraic curve defined over the finite field $\mathbb F\_q$.This bound enables us to provide explicit conditions on $q, g$ and $π$ for the non-existence of absolutely irreducible projective algebraic curves defined over $\mathbb F\_q$ of geometric genus $g$, arithmetic genus $π$ and with $N\_q(g)+π-g$ rational points.Moreover, for $q$ a square, we study the set of pairs $(g,π)$ for which there exists a maximal absolutely irreducible projective algeb…
The Picard number of a Fano manifold X obtained by blowing up a curve in a smooth projective variety is known to be at most 5, in any dimension greater than or equal to 4. We show that the Picard number attains to the maximal if and only if X is the blow-up of the projective space whose center consists of two points, the strict transform of the line joining them and a linear subspace or a quadric of codimension 2. This result is obtained as a consequence of a classification of special types of Fano manifolds.
It is known for a long time that a nonsingular real algebraic curve of degree 2k in the projective plane cannot have more than 7/2*k^2-9/4*k+3/2$ even ovals. We show here that this upper bound is asymptotically sharp, that is to say we construct a family of curves of degree 2k such that p/k^2 tends to 7/4$ as k tends to infinity, where p is the number of even ovals of the curves. We also show that the same kind of result is valid dealing with odd ovals.
We determine the zeta functions of trinomial curves in terms of Gauss sums and Jacobi sums, and we obtain an explicit formula of the genus of a trinomial curve over a finite field, then we study the conditions for a trinomial curve to be a maximal curve over a finite field.
Let $f: X \to B$ be a relatively minimal fibration of maximal Albanese dimension from a variety $X$ of dimension $n \ge 2$ to a curve $B$ defined over an algebraically closed field of characteristic zero. We prove that $K_{X/B}^n \ge 2n! χ_f$, which was conjectured by Barja in [2]. Via the strategy outlined in [5], it also leads to a new proof of the Severi inequality for varieties of maximal Albanese dimension. Moreover, when the equality holds and $χ_f > 0$, we prove that the general fiber $F$ of $f$ has to satisfy the Severi equality that $K_F^{n-1} = 2(n-1)! χ(F, ω_F)$. We also prove some …
Let $E$ be a rank 2, degree $d$ vector bundle over a genus $g$ curve $C$. The loci of stable pairs on $E$ in class $2[C]$ fixed by the scaling action are expressed as products of $\Quot$ schemes. Using virtual localization, the stable pairs invariants of $E$ are related to the virtual intersection theory of $\Quot E$. The latter theory is extensively discussed for an $E$ of arbitrary rank; the tautological ring of $\Quot E$ is defined and is computed on the locus parameterizing rank one subsheaves. In case $E$ has rank 2, $d$ and $g$ have opposite parity, and $E$ is sufficiently generic, it is…
Let C be a reduced, irreducible, not degenerate curve, not contained on surfaces of degree <s; when d=deg(C) is large with respect to s, the arithmetic genus p_a(c) is bounded by a function G(d, r, s) which is of type d^2/2s+O(d). The existence of such a bound for curves in P^3 was announced by Halphen in 1870 and proved by Gruson and Peskine in 1978; for curves in P^r, r>3, the bound is stated and proved by Chiantini, Ciliberto and Di Gennaro in 1993. The bound is sharp, at least for d sufficiently large (Chiantini-Ciliberto- Di Gennaro give examples of extremal curves for d large, that are i…
