On the genus of a maximal curve

By Gábor Korchmáros, F. Torres. 2000

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Abstract. The upper limit and the first gap in the spectrum of genera of $ \mathbf F_{q^2} $-maximal curves are known, see [34], [16], [35]. In this paper we determine the second gap. Both the first and second gaps are approximately constant times $q^2$, but this does not hold true for the third gap which is just 1 for $q\equiv 2{\pmod 3},$ while (at most) constant times q for $q\equiv 0{\pmod 3}.$ This suggests that the problem of determining the third gap which is the object of current work on $\mathbf F_{q^2}$-maximal curves could be intricate. Here, we investigate a relevant related problem namely that of characterising those $\mathbf F_{q^2}$-maximal curves whose genus is equal to the third (or possible the forth) largest value in the spectrum. Our results also provide some new evidence on $\mathbf F_{q^2}$-maximal curves in connection with Castelnuovo's genus bound, Halphen's theorem, and extremal curves.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/86934854e40f197d9b2de210292e8a5cf32bfdbe