The Weierstrass Point Theory: its connections between curves with many points, algebraic geometric Goppa codes, algebraic geometry over finite fields and finite geometry. In several topics of our research, the key reference is the Stoehr-Voloch paper (Zbl0593.14020) which has to do with a geometric approach to the Hasse-Weil upper bound and some classical results of curves such as Castelnuovo's genus bound.
(Numerical) Semigroups: On a Hurwitz's question
(1892) concerning the existence of semigroups that are not Weierstrass.
Working this question via Graph Theory: Ishii's paper (Zbl0593.1402), Kelarev and Quinn paper
(Zbl1005.20043) and Baker's paper (Zbl0552.93404).
Computations of the order bound for some specific type of semigroups which generalize for example those introduced by Bras-Amorós.
Curves with Many Points: For $q$ and $g$ fixed the function $N_q(g)$, the maximum number of $F_q$-rational points that a curve over $F_q$ of genus $g$ can have, is investigated. (In van der Geer and van der Vlugt tables we find pairs $(q,g)$, with $q$ and $g$ small, such that $N_q(g)$ is ``large".) A curve attaining the Hasse-Weil upper bound is called maximal. Two problems arise:
Finite Geometries: The existence of configurations of points in projective planes over finite fields (e.g. arcs, dense sets ...) arising from plane curves [The use of curves having 'pathological geometric behavior´ seems to be quite useful in studying this problem.]
Topics on Algebraic Geometric Goppa Codes (GG-codes):
Prym varities and Weierstrass points