Given an algebraic curve C of genus g, we say a point P is a Weierstrass point if dim(L(gP))>1, where L(D) is the Riemann-Roch space associated to a divisor D. One can generalize this to define a higher-order Weierstrass point, and one can also talk about the weight of a Weierstrass point.
In this talk, we will consider Weierstrass points on superelliptic curves, which are curves of the form y^n=f(x) for f(x) a polynomial of degree d. Under a mild hypothesis, it is well known that the branch points of such curves are Weierstrass points. We will investigate the weights of these branch points for given values of n and d and use them to obtain some asymptotic results. We will also look at the various weight distributions that can arise on certain families of hyperelliptic curves of low genus and prescribed automorphism groups.
Ordinary and higher-order Weierstrass points on cyclic covers of the projective line
Nov. 04, 2014 3pm (Math 350)
Algebraic Geometry
Pál Zsámboki (University of Washington)
X
Starting with linear endomorphisms of a 2-dimensional complex vector space, and moving on to line bundles on a real conic, I will discuss how the problem of finding a moduli space can involve having to extend our notion of geometrical space. After that, I will talk about how a choice of a meaningful compactification of the moduli space of vector bundles on a surface has led to the proof of irreducibility of said moduli space.
I will discuss an approach to getting a meaningful boundary to the moduli space of G-bundles on a smooth, proper scheme using derived objects. Homotopical Algebraic Geometry will arise as the natural framework to treat this problem in. No knowledge of higher stacks will be assumed.
This will be report on a series of joint papers with Hang Wang (U. Adelaide, Australia). The general aim of these papers is to use tools from noncommutative geometry to study conformal geometry and some noncommutative incarnations of conformal geometry. In this talk, I will present two main results. The first main result is a reformulation of the local index formula in the setting of conformal-diffeomorphism invariant geometry. The second main result is the construction of a new family of global conformal invariants taking into account the action of the group of conformal-diffeomorphisms. These invariants are not of the same type as the conformal invariants considered by Spyros Alexakis in his celebrated solution of the Deser-Schwimmer. The arguments leading to both results heavily rely on noncommutative geometry. In particular, a crucial use is made of the conformal invariance of the Connes-Chern character. However, the main results are ultimately stated in differential-geometric fashion.
Noncommutative geometry and conformal geometry: local index formula and conformal invariants.
Ordinary and higher-order Weierstrass points on cyclic covers of the projective line