Optimal and maximal singular curves

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

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