Crotaphytus reticulatus

Number of Pages: 2Integrative BiologyGeological Science

A new family of maximal curves

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--Korchm\'aros 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, by considering some Galois subcovers of $\mathcal X_n$

On Hyperfocused Arcs in PG(2,q)

A k-arc in a Dearguesian projective plane whose secants meet some external line in k-1 points is said to be hyperfocused. Hyperfocused arcs are investigated in connection with a secret sharing scheme based on geometry due to Simmons. In this paper it is shown that point orbits under suitable groups of elations are hyperfocused arcs with the significant property of being contained neither in a hyperoval, nor in a proper subplane. Also, the concept of generalized hyperfocused arc, i.e. an arc whose secants admit a blocking set of minimum size, is introduced: a construction method is provided, together with the classification for size up to 10

On the spectrum of genera of quotients of the Hermitian curve

We investigate the genera of quotient curves $\mathcal H_q/G$ of the $\mathbb F_{q^2}$-maximal Hermitian curve $\mathcal H_q$, where $G$ is contained in the maximal subgroup $\mathcal M_q\leq{\rm Aut}(\mathcal H_q)$ fixing a pole-polar pair $(P,\ell)$ with respect to the unitary polarity associated with $\mathcal H_q$. To this aim, a geometric and group-theoretical description of $\mathcal M_q$ is given. The genera of some other quotients $\mathcal H_q/G$ with $G\not\leq\mathcal M_q$ are also computed. Thus we obtain new values in the spectrum of genera of $\mathbb F_{q^2}$-maximal curves. A plane model for $\mathcal H_q/G$ is obtained when $G$ is cyclic of order $p\cdot d$, with $d$ a divisor of $q+1$