Narrow Results

We show that the log canonical bundle, κ, of is very ample, show the homogeneous coordinate ring is Koszul, and give a nice set of rank 4 quadratic generators for the homogeneous ideal: The embedding is equivariant for the symmetric group, and the image lies on many Segre embedded copies of ℙ¹ × ⋯ × ℙ^n-3, permuted by the symmetric group. The homogeneous ideal of is the sum of the homogeneous ideals of these Segre embeddings.

It is known that orbit reduction can be performed in one or two stages and it has been proven that the two processes are symplectically equivalent. In the context of orbit reduction by one stage, we shall write an expression for the reduced two-form in the general case and obtain the equations of motion derived from this theory. Then we shall develop the same process in the case in which the symmetry group has a normal subgroup to get the reduced symplectic form by two stages and the consequent orbit reduced equations. In both cases, we shall illustrate the method with three physical examples.