Optimal and maximal singular curves

Abstract

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 $\pi$ for the non-existence of absolutely irreducible projective algebraic curves defined over $\mathbb{F}\_q$ of geometric genus $g$, arithmetic genus $\pi$ and with $N\_q(g)+\pi -g$ rational points.Moreover, for $q$ a square, we study the set of pairs $(g,\pi )$ for which there exists a maximal absolutely irreducible projective algebraic curve defined over $\mathbb{F}\_q$ of geometric genus $g$ and arithmetic genus $\pi$, i.e. with $q+1+2g\sqrt{q}+\pi -g$ rational points.