By Robin Hartshorne, C. Musili
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Additional resources for Ample Subvarieties of Algebraic Varieties
Put algebra of observables of the -swarm of orbits of the -action. Recall, see , , and that the classical quotient scheme (exist and) is isomorphic §8, that to . Let be the ﬁnite family of irreducible (simple) -representations. e. such that , and use Theorem 14. It says that there exists and a morphism a scheme extending the natural morphism If the scheme has to be empty, since any with support outside the origin in , correspond to a reduced orbit, and so necessarily is a simple have length less or equal to the order of , and any with support in -representation with trivial -action, so .
E. the action of on using the -linear and -invariant section of the morphism , induced by ﬁxing a -basis for , O. Laudal / The Structure of Simp<∞ (A) mapping, multiplicatively, We obtain: where , subject to the relation in with the if to -action given by , and Moreover, the (1,1)-term of the matrix for looks like and the (1,2)-term has the form The (1,1)-term of the matrix, has the form and the (1,2)-term looks like , and 37 to , see §8 of . 38 O. Laudal / The Structure of Simp<∞ (A) Here is the image of in .
In this case , and , and is the well known singularity. Clearly has two simple (irreducible) representations of dimension 1, , where acts as , respectively, and the regular representation, is the sum of these. The orbits of in , are either of length 2, corresponding to a simple -module of dimension 2, or are reduced to the origin. Therefore the indecomposable -modules of dimension 2, must all have support at the origin. They must therefore be given by the indecomposables of representation graph, / Now all such are given in terms of the following actions of on the vectorspace .
Ample Subvarieties of Algebraic Varieties by Robin Hartshorne, C. Musili