On fixed points of algebraic actions on \(\mathbb{C}^n\) (Q1576932)

Consider the natural representation of \(G := \text{ SL}_2({\mathbb C})\) on the vector space \(V\) of homogeneous forms of degree \(8\) in the variables \(x\) and \(y\). Let \(W \subset V\) be the subspace generated by \(x^iy^{8-i}\) where \(0 \leq i \leq 2\) and \(6 \leq i \leq 8\). The authors show that \(L := G \cdot (W + x^4y^4)\) is not locally closed in \(V\). This ends by a negative answer to a discussion wether \(L\) could serve as an example of an affine space with a fixed point free reductive group action. Moreover, the authors show that no open subset of the open kernel of \(L\) is isomorphic to an affine space.