Explicit Solution By Radicals, Gonal Maps and Plane Models of Algebraic Curves of Genus 5 or 6

We give explicit computational algorithms to construct minimal degree (always $\le 4$) ramified covers of \Prj^1 for algebraic curves of genus 5 and 6. This completes the work of Schicho and Sevilla (who dealt with the $g \le 4$ case) on constructing radical parametrisations of arbitrary genus $g$ curves. Zariski showed that this is impossible for the general curve of genus $\ge 7$. We also construct minimal degree birational plane models and show how the existence of degree 6 plane models for genus 6 curves is related to the gonality and geometric type of a certain auxiliary surface.Comment: v3: full version of the pape

Rational plane curves parameterizable by conics

We introduce the class of rational plane curves parameterizable by conics as an extension of the family of curves parameterizable by lines (also known as monoid curves). We show that they are the image of monoid curves via suitable quadratic transformations in projective plane. We also describe all the possible proper parameterizations of them, and a set of minimal generators of the Rees Algebra associated to these parameterizations, extending well-known results for curves parameterizable by lines.Comment: 28 pages, 1 figure. Revised version. Accepted for publication in Journal of Algebr

Complete intersections in simplicial toric varieties

Given a set $\mathcal A = \{a_1,\ldots,a_n\} \subset \mathbb{N}^m$ of nonzero vectors defining a simplicial toric ideal $I_{\mathcal A} \subset k[x_1,...,x_n]$, where $k$ is an arbitrary field, we provide an algorithm for checking whether $I_{\mathcal A}$ is a complete intersection. This algorithm does not require the explicit computation of a minimal set of generators of $I_{\mathcal A}$. The algorithm is based on the application of some new results concerning toric ideals to the simplicial case. For homogenous simplicial toric ideals, we provide a simpler version of this algorithm. Moreover, when $k$ is an algebraically closed field, we list all ideal-theoretic complete intersection simplicial projective toric varieties that are either smooth or have one singular point.Comment: 28 pages, 2 tables. To appear in Journal of Symbolic Computatio

Point counting on curves using a gonality preserving lift

We study the problem of lifting curves from finite fields to number fields in a genus and gonality preserving way. More precisely, we sketch how this can be done efficiently for curves of gonality at most four, with an in-depth treatment of curves of genus at most five over finite fields of odd characteristic, including an implementation in Magma. We then use such a lift as input to an algorithm due to the second author for computing zeta functions of curves over finite fields using $p$-adic cohomology