The Picard Group of Brauer-Severi Varieties

In this note we provide explicit generators of the Picard groups of cyclic Brauer-Severi varieties defined over the base field. In particular, for all Brauer-Severi surfaces. To produce these generators we use the Twisting Theory for smooth plane curves

A note on the stratification by automorphisms of smooth plane curves of genus 6

In this note, we give a so-called representative classification for the strata by automorphism group of smooth $\bar{k}$-plane curves of genus $6$, where $\bar{k}$ is a fixed separable closure of a field $k$ of characteristic $p = 0$ or $p > 13$. We start with a classification already obtained by the first author and we use standard techniques. Interestingly, in the way to get these families for the different strata, we find two remarkable phenomenons that did not appear before. One is the existence of a non $0$-dimensional final stratum of plane curves. At a first sight it may sound odd, but we will see that this is a normal situation for higher degrees and we will give a explanation for it. We explicitly describe representative families for all strata, except for the stratum with automorphism group $\mathbb{Z}/5\mathbb{Z}$. Here we find the second difference with the lower genus cases where the previous techniques do not fully work. Fortunately, we are still able to prove the existence of such family by applying a version of Luroth's theorem in dimension $2$

Twists of Elliptic Curves

In this note we extend the theory of twists of elliptic curves as presented in various standard texts for characteristic not equal to two or three to the remaining characteristics. For this, we make explicit use of the correspondence between the twists and the Galois cohomology set $H^1\big(\operatorname{G}_{\overline{K}/K}, \operatorname{Aut}_{\overline{K}}(E)\big)$. The results are illustrated by examples

Primes dividing invariants of CM Picard curves

We give a bound on the primes dividing the denominators of invariants of Picard curves of genus 3 with complex multiplication. Unlike earlier bounds in genus 2 and 3, our bound is based not on bad reduction of curves, but on a very explicit type of good reduction. This approach simultaneously yields a simplification of the proof, and much sharper bounds. In fact, unlike all previous bounds for genus 3, our bound is sharp enough for use in explicit constructions of Picard curves

A bound on the primes of bad reduction for CM curves of genus 3

We give bounds on the primes of geometric bad reduction for curves of genus three of primitive CM type in terms of the CM orders. In the case of elliptic curves, there are no primes of geometric bad reduction because CM elliptic curves are CM abelian varieties, which have potential good reduction everywhere. However, for genus at least two, the curve can have bad reduction at a prime although the Jacobian has good reduction. Goren and Lauter gave the first bound in the case of genus two. In the cases of hyperelliptic and Picard curves, our results imply bounds on primes appearing in the denominators of invariants and class polynomials, which are important for algorithmic construction of curves with given characteristic polynomials over finite fields

On quadratic progression sequences on smooth plane curves

We study the arithmetic (geometric) progressions in the $x$-coordinates of quadratic points on smooth projective planar curves defined over a number field $k$. Unless the curve is hyperelliptic, we prove that these progressions must be finite. We, moreover, show that the arithmetic gonality of the curve determines the infinitude of these progressions in the set of $\overline{k}$-points with field of definition of degree at most $n$, $n\ge 3$

Introducción: El folklore en Canarias

Número dedicado a: Islas de San Miguel de la Palma y La Gomer