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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) fordoi:10.1090/conm/686/13776 fatcat:rbffqwa63zh2pbvk3o4x52gxma