Brady Ali Medina, Department of Pure Mathematics, University of Waterloo
"A different way to generalize the Weierstrass semigroup"
We propose a different way to generalize the concept of a Weierstrass semigroup H_P associated to a point P in a curve X of genus g. We start by defining the Weierstrass set of a vector bundle F with respect to a point P, and we prove that this set is an H_P - relative ideal. Also, if the vector bundle is semistable, we prove that the largest gap is less than 2g − μ(F), where μ(F) denotes the slope of F. Furthermore, when F is a line bundle, we define the Weierstrass semigroup of F with respect to a divisor D and we find that the largest gap is less than 2g − deg(F)/ deg(D). Moreover, in the case when D = P we find that the cardinality of the set of gaps is exactly l(K_X − deg(F)), which is a theorem analogous to the Weierstrass Gap Theorem.
Zoom meeting: https://zoom.us/j/93859138328