TITLE: Reference for an algebraic group preserving a cubic form
QUESTION [6 upvotes]: Let $R=k[u,v,w]$ and $p\in R$ be a cubic form. Let $G$ be the group of graded automorphisms of $R$ which preserve $p$, i.e., $G$ is the subgroup
of $GL_3(k)$ consisting of elements $g$ such that $g(p) \in k p$. My question: is $G$ some well known algebraic group?

REPLY [3 votes]: If the field is $\mathbb{C}$ (or algebraically closed of characteristic $\not=2,3$), then you can put any smooth cubic into the Hessian form:
$$X^3+Y^3+Z^3+\lambda XYZ=0$$
for some $\lambda\in \mathbb{C}$. This corresponds to put the nine inflection onto the intersection of the curve with $XYZ=0$.
Then, the pencil generated by the cubic and $XYZ$ is the Hessian pencil (which corresponds to let $\lambda$ vary in the equation) and is preserved by the classical Hessian group $G\subset \mathrm{PGL}(3,\mathbb{C})$, of order $216$. The groups acts on the set of parameters, paramertised by $\mathbb{P}^1$, and the kernel of this action contains the group $(\mathbb{Z}/3\mathbb{Z})^3\rtimes \mathbb{Z}/2\mathbb{Z}$ generated by the diagonal action $$[X:Y:Z]\mapsto [X:\theta Y:\theta^2Z]$$
$\theta^3=1$ and the group of permutations. This shows that you have generically this group as a group of automorphisms of $\mathbb{P}^2$  preserving the curve. For some values of $\lambda$, you have other automorphisms. Indeed, each of the elements of the Hessian group has two fixed points on $\mathbb{P}^1$.
I suggest to read the nice article "The Hesse pencil of plane cubic curves" of M. Artebani and I. Dolgachev on this subject http://arxiv.org/abs/math/0611590 .