Let $f\in\mathbb{C}[x,y,z]_4$ be a smooth ternary quartic. And let $G$ be the automorphism group of $f$ (those elements in $GL_3(\mathbb{C})$ fixing the curve $f=0$). For example in Dolgachev's book (http://www.math.lsa.umich.edu/~idolga/CAG.pdf) one can find a list of all automorphism groups that can appear.
My question is: What's the automorphism group of a generic $f$? Is this group trivial? Or are the sets with fixed automorphism group inside $\mathbb{C}[x,y,z]_4$ not Zariski-closed and the question doesnt make too much sense?
Thx in advance
