Student Algebraic Geometry

When studying an algebraic curve X of genus at least 3, it often helps to consider (1) an embedding of X into some ambient space, and (2) some set of "reference" points on X. One natural choice for (1) is the canonical embedding X->P^r into projective space (where r = g-1) and for (2) we choose points that are "special" with respect to the set of hyperplanes in P^r -- these are called Weierstrass points of X. A second natural choice for (1) is the Abel-Jacobi embedding X->Jac(X) into the Jacobian variety, and for (2) we choose the set of N-torsion points which happen to lie on the curve.

Depending on the curve X, the set of points you get from these two methods may or may not overlap. However, a recent result of Girard, Kohel, and Ritzenthaler (2005) proves that for a generic curve, the points you get in these two ways are entirely disjoint. In this talk I will discuss the proof of this result. Some familiarity with algebraic curves will be assumed. Speaker(s): Harry Richman (UM)