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.
Domains
Algebraic Geometry [math.AG]
Origin : Files produced by the author(s)
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