Linear \(G_a\) actions on affine spaces and associated rings of invariants (Q1239204)

Let \(k\) be an algebraically closed field and \(G_a\) the additive group scheme over \(k\). Assume that char \(k=0\). Then any linear \(G_a\)-action is a direct sum of basic actions. The last ones can be described as follows. Let \(V\) be a space of dimension 2 and \(\rho:G_a\to GL(V)\) be the map \(\rho(t)=\left(\begin{smallmatrix} 1 & t\\ 0 & 1\end{smallmatrix}\right)\).Take \(W=s^{n-1}V\) (\(n-1\)-th symmetric power) and consider the action of \(G_a\) on the affine space \(W\). For these ``basic'' actions, the author computes a ring of invariants in an explicit way through generators. The numbers of generators is \(2n-4\). This gives another proof of the Weitzenböck theorem for linear \(G_a\)-actions. The method gives also a lot of information about the ring of invariants. If 11.3 it is a polynomial ring and in general case it is a factorial complete intersection. In the whole scheme where \(G_a\) acts, there exists an invariant hypersurface whose complement is \(G_a\)-stable and possesses a geometrical quotient under the action of \(G_a\).