This award supports collaborative research on algebraic geometry. Robert Lax of Louisiana State University is an expert on Weierstrass points on complex Gorenstein curves, while Arnaldo Garcia of IMPA (Rio de Janeiro) is an expert on curves in positive characteristic. The notions of classical and non classical curves will be extended to singular curves. Questions concerning classicality of curves and normalization and classicality of curves and degenerations will be considered. On smooth curves in characteristic zero, the investigators will study possible weights and gap sequences of Weierstrass points on curves that are two-sheeted covers of other curves. On smooth curves in positive characteristic, the investigators will study relation between the defining equation and the classicality of the curve. Finally, on curves over finite fields, the investigators will study relations between the Weierstrass gap sequence at a point and geometric Goppa codes defined using a multiple of that point. The question of the number of rational points on curves with certain types of Weierstrass points will also be considered. While this research is sure mathematics, there exists the potential for important application in coding theory due to the connections with Goppa codes.
