arXiv subjects

Saeed Tafazolian

Publications and source records attributed to Saeed Tafazolian.

18 recordsLinked to original sources

arXiv2026

Transitive automorphism groups of maximal curves

Let $q=p^h$ be an odd prime power and let $X/\F_{q^2}$ be a maximal curve of genus at least two. We classify the curves for which the full geometric automorphism group is transitive on the set of $\F_{q^2}$-rational points. We prove that, if $h>1$, then $X$ is the Hermitian curve. If $h=1$, the only additional possibility occurs for $q=5$: the unique $\F_{25}$-maximal genus-three curve, namely the maximal $S_4$-model of the Klein quartic. The proof separates tame and wild actions. In the tame case, genus bounds, signatures of quotient maps, specialization to characteristic zero, low-genus automorphism classifications, and a Hasse--Witt computation reduce the problem to the Klein quartic. In the wild case, the rational points are identified with the Sylow $p$-subgroups of the automorphism group. Noncyclic Sylow subgroups are treated using a finite-group classification theorem together with Henn's large-automorphism classification, while cyclic Sylow subgroups are excluded by local ramification and the Riemann--Hurwitz formula. Combined with the known characteristic-two result, this yields the classification for all prime powers.

math.AG↗

arXiv2026

Nakajima-extremal Artin--Schreier covers of ordinary elliptic curves in characteristic $2$

Let $k$ be an algebraically closed field of characteristic $2$ and let $q=2^h$, $h\ge3$. We construct ordinary bielliptic curves $X$ of genus $q+1$ for which \[ \Aut(X)\cong \Dih(C_q)\times C_2, \qquad |\Aut(X)|=4q=4(g(X)-1). \] These curves realize case \textup{(ib)} in the classification of Giulietti--Korchmáros and give an infinite family answering a problem posed by Korchmáros. More generally, we prove that every curve in case \textup{(ib)} arises from the same construction. Case \textup{(ib)} occurs exactly in genera $g=2^h+1$ with $h\ge2$. For $g\ge9$ we determine the full automorphism group; in genus $5$ we determine its Sylow $2$-subgroup but do not claim the full automorphism group. For every fixed $q\ge8$, the isomorphism classes in case \textup{(ib)} of genus $q+1$ are parametrized bijectively by $(k^\times)^2$. The construction is described in terms of an ordinary elliptic curve and an invariant differential. We determine the short orbits and ramification, the unramified cyclic quotients, and the quotients by the central involutions. For $q=8$ an explicit plane model over $\F_2$ is given.

math.AG↗

arXiv2026

On Skabelund's Ray Class Field Covers of the Suzuki and Ree Curves

Let $\Sm_q$ and $\Rm_q$ denote the Suzuki and Ree curves. Motivated by the Giulietti--Korchmáros curve, Skabelund constructed cyclic covers $\tSm_q$ and $\tRm_q$ of these curves and proved that they are maximal over $\F_{q^4}$ and $\F_{q^6}$, respectively. In the same paper he associated to the Suzuki and Ree curves certain ray class field covers $\Sm_{\rm rcf}$ and $\Rm_{\rm rcf}$, and showed that there are towers \[ \Sm_{\rm rcf}\longrightarrow\tSm_q\longrightarrow\Sm_q, \qquad \Rm_{\rm rcf}\longrightarrow\tRm_q\longrightarrow\Rm_q . \] Computations for small values of $q$ suggested that the first map in each tower is always an isomorphism, and the general case was left open. We prove that \[ \Sm_{\rm rcf}=\tSm_q,\qquad \Rm_{\rm rcf}=\tRm_q \] for every admissible $q$, the comparison being made over $\F_{q^4}$ in the Suzuki case and over $\F_{q^6}$ in the Ree case. The proof combines a Kummer normal form of the ray class extension, allowing a constant twist, with the centrality of its Galois group among lifted automorphisms, the standard involution of the base curve, and the first positive non-gap at the rational point at infinity.

math.AG↗

arXiv2026

Maximal and minimal curves of the form $y^3=x^{(q^2+1)/2}+x$

Let $p\ge 5$ be a prime with $p\equiv -1\pmod 3$, let $q=p^r$, and consider \[ \cC:\qquad y^3=x^{(q^2+1)/2}+x \] over $\F_{q^6}$. We prove the exact formula \[ \#\cC(\F_{q^6})=q^6+1+(-1)^{r+1}(q^2-1)q^3. \] Since $g(\cC)=(q^2-1)/2$, the curve is maximal when $r$ is odd and minimal when $r$ is even. The proof uses a birational Kummer model and an explicit Jacobi-sum point count. A congruence together with Frobenius invariance reduces the relevant Jacobi sums to cubic Gauss sums, whose sign is determined from the Fermat cubic. In particular, the maximality of $y^3=x^{13}+x$ over $\F_{5^6}$ appears as the first case of an infinite family.

math.NT↗

arXiv2026

Explicit LCP of MDS Codes and LCD Codes on Hyperelliptic Curves via Mumford Representation

We study algebraic geometry codes on hyperelliptic curves of genus $g \geq 2$ with complementarity properties. Our first contribution is a characterization of non-special divisors of degree $g$ and $g-1$ via the polynomial degrees of their reduced Mumford representation, reducing a classical hard geometric problem to a single-degree test on univariate polynomials. Using this, we construct Linear Complementary Pairs (LCP) of codes via polynomial arithmetic on the Jacobian and provide a criterion in terms of Mumford degrees for the resulting codes to be Maximum Distance Separable (MDS). Under a $2$-torsion condition in the Jacobian, equivalently a divisibility condition on the Mumford polynomials, we obtain explicit multipliers that turn these pairs into Linear Complementary Dual (LCD) codes. Finally, we apply this framework to the maximal hyperelliptic curve $\mathcal{X} \colon y^2 = x^q + x$ over $\mathbb{F}_{q^2}$ and give explicit examples of MDS LCD codes with parameters $[2q,q,q+1]_{q^2}$ for $q = 4, 5, 7$, verified computationally; we conjecture, with heuristic support, that such codes exist for all $q \geq 4$.

math.AG↗

arXiv2026

Jacobian algebras and variation of hyperplane sections

We study the variation in moduli of hyperplane sections of a hypersurface $V(f)\subseteq \mathbf P^n$ with at most isolated singularities. Using the Milnor algebra $M(f)$, we give an infinitesimal quotient criterion for the hyperplane-section map $Φ(f):(\mathbf P^n)^*\dashrightarrow M(d,n-1)$ to be generically finite onto its image. The passage from the infinitesimal quotient to the coarse moduli space is justified by a local GIT slice argument. Our approach gives a Jacobian-algebraic extension of the Beauville--Patel--Riedl--Tseng theory from smooth hypersurfaces to hypersurfaces with isolated singularities. In the smooth case it recovers the Lefschetz criterion and, using recent weak Lefschetz results, gives generic finiteness for $n\geq 3$ in the range $d\geq n+2$. In the singular case a new obstruction appears: a linear Jacobian syzygy, equivalently, for non-cones, a positive-dimensional projective automorphism group. After this obstruction is excluded, maximal infinitesimal variation is governed by the injectivity of the critical Lefschetz map $\ell:M(f)_{d-1}\to M(f)_d$. We apply the criterion to plane curves, surfaces in $\mathbf P^3$, and hypersurfaces admitting singular hyperplane sections, obtaining new criteria involving nodal sections and an application to the Schoen quintic threefold.

math.AG↗

arXiv2026

On the Maximality, Weierstrass Semigroups, and Automorphism Group of the Curve $Y^{q+1} = X^n(X^n + 1)$

We study the algebraic curve over $\mathbb{F}_{q^2}$ defined by $y^{q+1} = x^n(x^n+1)$, where $n$ is a positive integer coprime to the characteristic. We first prove (when $q$ is odd) that the nonsingular model of this curve is $\mathbb{F}_{q^2}$-maximal if and only if $n \mid (q+1)$. Writing $n = \frac{q+1}{m}$, we obtain a family of maximal curves parameterized by the divisors $m$ of $q+1$, which extends the previously studied case $m=3$ corresponding to maximal curves with the third largest possible genus. For this family, we determine the Weierstrass semigroups at several classes of rational points, including those lying above the branch points of the natural projection. These semigroups are described explicitly in terms of $q$ and $m$, and exhibit different behaviors depending on the arithmetic properties of $m$. Moreover, we determine the full automorphism group of the curve under a mild condition on the characteristic. Our results extend an earlier work on the case $m=3$ and provide new insight into the structure of this family of maximal curves.

math.AG↗

arXiv2026

Construction of Non-special Divisors on Kummer Covers With Arbritary Ramification For LCP Codes

Linear Complementary Pairs (LCP) of algebraic geometry (AG) codes offer strong resistance against side-channel and fault-injection attacks, but their construction depends critically on the explicit identification of non-special divisors of degree $g$ and $g-1$. Existing constructions are restricted to Kummer extensions where divisors are supported exclusively on totally ramified places, significantly limiting the range of applicable function fields and codes. We remove this restriction by developing a framework for general Kummer extensions $y^m = \prod_{i=1}^r (x-α_i)^{λ_i}$ over finite fields with arbitrary ramification. Using Galois group actions and invariant divisor techniques, we establish necessary and sufficient conditions for non-speciality with no constraint on the support, yielding explicit constructions where previous methods fail. Our approach replaces the computationally intensive Weierstrass semigroup machinery with a more direct and efficient framework. As an application, we construct new explicit families of LCP AG codes with determined parameters $[n,k,d]$, covering three ramification regimes. The resulting codes meet or approach the Goppa designed distance, offering greater flexibility for cryptographic applications.

math.AG↗

arXiv2026

Veronese Avoiding Hypersurfaces

We introduce Veronese-Avoiding hypersurfaces, inspired by the theory of associated forms of Alper--Isaev. In the smooth case, we reinterpret their criterion via Macaulay inverse systems: the Veronese-Avoiding condition is equivalent to the non-degeneracy of the associated form. In the singular case, our main theorem shows that a reduced hypersurface with exactly $n$ isolated singular points is Veronese-Avoiding if and only if these points are ordinary nodes in general linear position; we also classify singular plane cubics and treat fewer than $n$ nodes via a natural rational map. We then study the parameter space, proving local closedness and identifying a distinguished irreducible nodal locus. Finally, we prove a Lefschetz-type consequence for the Milnor algebra in degree $1$.

math.AG↗

arXiv2026

Automorphisms of Plane Curves defined from Chebychev polynomials

We investigate the automorphism groups of the algebraic curves \[ \mathcal{C}_d : y^d = φ_d(x), \] where $φ_d(x)$ denotes the Chebyshev polynomial of degree $d$, defined over a field $k$ with $p:=\operatorname{char}(k) \nmid 2d$. We determine the full automorphism group of $\mathcal{C}_d$ in all the cases considered in this paper, namely for $d=4$, and more generally when $2d = p^r+1$ or $4d = p^r+1$. For all other $d>4$, Expectation~\ref{3.19} predicts what the automorphism group should be. As an application, we show that certain maximal curves of the same genus are not isomorphic.

math.AG↗

arXiv2026

Maximal Subcovers of the Skabelund Curve: Uniqueness via Genus and Automorphism Groups

We establish a rigidity phenomenon for a family of intermediate covers of the Skabelund curve over $\mathbb{F}_{q^4}$. The Skabelund curve, introduced by D.~Skabelund as a cyclic cover of the Suzuki curve, is a maximal curve with a large automorphism group and plays a central role in the theory of maximal curves over finite fields. For the intermediate covers arising from this construction, we determine their full automorphism groups and compute the Weierstrass semigroups at all $\mathbb{F}_{q^4}$-rational points. Using these structural and arithmetic invariants, we prove that each curve in the family is uniquely determined, up to isomorphism over its field of definition, by the pair consisting of its genus and its full automorphism group. This provides a rigidity-type classification of intermediate Suzuki-type covers; in particular, the Skabelund curve itself is uniquely characterized within this family by its genus and automorphism group.

math.AG↗

arXiv2025

Explicit Families of Hyperelliptic Curves with CM Jacobians

We construct explicit families of hyperelliptic curves over $\QQ$ whose Jacobians admit complex multiplication (CM). Each curve in these families is defined by \[ v^2 = (u+2)\,φ_d(u), \quad d = 2^e \text{ or } d=p \geq 3 \text{ prime}, \] where $φ_d(x)$ is the Chebyshev polynomial of degree $d$. We prove that the Jacobians are simple and determine the associated CM-fields explicitly. Our approach exploits the interplay between Chebyshev polynomials and Galois coverings, providing concrete examples of abelian varieties with CM and explicit criteria for their construction.

math.AG↗

arXiv2025

Locally recoverable codes with multiple recovering sets from maximal curves

In this paper, we present a construction of locally recoverable codes (LRCs) with multiple recovery sets using algebraic curves with many rational points. By leveraging separable morphisms between smooth projective curves and expanding the class of curves previously considered, we significantly generalize and enhance the framework. Our approach corrects certain inaccuracies in the existing literature while extending results to a broader range of curves, thereby achieving better parameters and wider applicability. In addition, the constructions presented here result in LRCs with large availability.

math.AG↗

arXiv2022

Hankel edge ideals of trees and (semi-)Hamiltonian graphs

In this paper, we study the Hankel edge ideals of graphs. We determine the minimal prime ideals of the Hankel edge ideal of labeled Hamiltonian and semi-Hamiltonian graphs, and we investigate radicality, being a complete intersection, almost complete intersection and set theoretic complete intersection for such graphs. We also consider the Hankel edge ideal of trees with a natural labeling, called rooted labeling. We characterize such trees whose Hankel edge ideal is a complete intersection, and moreover, we determine those whose initial ideal with respect to the reverse lexicographic order satisfies this property.

math.AC↗

arXiv2019

The $a$-number of Certain Hyperelliptic Curves

In this paper, we compute a formula for the $a$-number of certain hyperelliptic curves given by the equation $y^2= x^m+1$ for infinitely many values of $m$. The same question is studied for the curve corresponding to $y^2= x^m+x$.

math.AC↗

arXiv2019

On certain maximal hyperelliptic curves related to Chebyshev polynomials

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 $y^2=(x^2-4)φ_d(x)$.

math.AG↗

arXiv2016

On the spectrum for the genera of maximal curves over small fields

Motivated by previous computations in Garcia, Stichtenoth and Xing (2000) paper ,we discuss the spectrum $\mathbf{M}(q^2)$ for the genera of maximal curves over finite fields of order $q^2$ with $7\leq q\leq 16$. In particular, by using a result in Kudo and Harashita(2016) paper, the set $\mathbf{M}(7^2)$ is completely determined.

math.AG↗

arXiv2011

Covered by Lines and Conic Connected Varieties

We study some properties of an embedded variety covered by lines and give a numerical criterion ensuring the existence of a singular conic through two of its general points. We show that our criterion is sharp. Conic-connected, covered by lines, QEL, LQEL, prime Fano, defective, and dual defective varieties are closely related. We study some relations between the above mentioned classes of objects using celebrated results by Ein and Zak.

math.AG↗