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 Fq. This bound enables us to provide explicit conditions on q, g and π for the nonexistence of absolutely irreducible projective algebraic curves defined over Fq of geometric genus g, arithmetic genus π and with Nq(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 Fq of geometric genus g and arithmetic genus π, i.e. with q + 1 + 2g √ q + π − g rational points.