On Hermite's invariant for binary quintics

The Hermite invariant H is the defining equation for the hypersurface of binary quintics in involution. This paper analyses the geometry and invariant theory of H. We determine the singular locus of this hypersurface and show that it is a complete intersection of a linear covariant of quintics. The projective dual of this hypersurface can be identified with itself via an involution. It is shown that the Jacobian ideal of H is perfect of height two, and we describe its SL_2-equivariant minimal resolution. The last section develops a general formalism for evectants of covariants of binary forms, which is then used to calculate the evectant of H

Four lectures on secant varieties

This paper is based on the first author's lectures at the 2012 University of Regina Workshop "Connections Between Algebra and Geometry". Its aim is to provide an introduction to the theory of higher secant varieties and their applications. Several references and solved exercises are also included.Comment: Lectures notes to appear in PROMS (Springer Proceedings in Mathematics & Statistics), Springer/Birkhause

Symmetries of Riemann surfaces and magnetic monopoles

This thesis studies, broadly, the role of symmetry in elucidating structure. In particular, I investigate the role that automorphisms of algebraic curves play in three specific contexts; determining the orbits of theta characteristics, influencing the geometry of the highly-symmetric Bring’s curve, and in constructing magnetic monopole solutions. On theta characteristics, I show how to turn questions on the existence of invariant characteristics into questions of group cohomology, compute comprehensive tables of orbit decompositions for curves of genus 9 or less, and prove results on the existence of infinite families of curves with invariant characteristics. On Bring’s curve, I identify key points with geometric significance on the curve, completely determine the structure of the quotients by subgroups of automorphisms, finding new elliptic curves in the process, and identify the unique invariant theta characteristic on the curve. With respect to monopoles, I elucidate the role that the Hitchin conditions play in determining monopole spectral curves, the relation between these conditions and the automorphism group of the curve, and I develop the theory of computing Nahm data of symmetric monopoles. As such I classify all 3-monopoles whose Nahm data may be solved for in terms of elliptic functions

Computation of Affine Covariants of Quadratic Bivariate Differential Systems

International audienceThis paper deals with affine covariants of autonomous differential systems. We give a constructive method for computing them. This method allows in particular to deduce a (minimal) system of generators of the algebra of affine covariants from one of centro-afffine invariants.In the case of planar quadratic differential systems, we give a minimal system of the algebra of center-affine covariants and using the previous method we construct a minimal system of generators of the algebra of affine covariants. Computations are made with Maple. All algorithms constitute the package SIB