When studying GIT stability of hypersurfaces $d$ of $\mathbb P^n$ we look at the Hilbert Scheme $H=\mathbb P^N$ parametrizing homogeneous polynomials $f_d(x_0,\ldots,x_n)$ of degree $d$. There is normally a group $G$ acting on $\mathbb P^n$ which induces an action on $H$.

The GIT compactification is then $H^{ss}//G$ where we have restricted to semistable surfaces.

A natural option for $G$ would be to take $G=\mathrm{Aut}(\mathbb P^n)=PGL(n+1)$ so that we identify two hypersurfaces if they are related by an automorphism of $\mathbb P^n$. However in a few sources I have consulted this is never the case. Indeed, they always use either

$G=GL(n+1)$ (e.g. Mukai's "An Introduction to Invariants and Moduli" in the study of cubic surfaces)

$G=SL(n+1)$ (e.g. Alcock's "The Moduli Space of Cubic Threefolds")

**Q1**: Why is $PGL(n+1)$ never used?

I understand that any group used must include $SL(n+1)$ in order to apply Hilbert-Mumford's numerical criterion. So my second question is:

**Q2**: How to decide if choosing GL or SL? Doesn't using SL imply that we are considering two isomorphic hypersurfaces to be different when taking quotient?

**Q3** I understand (see comments) that $SL$ gives a unique linearization of the line bundle, but shouldn't $PGL$ be used in all cases when constructing compactifications of the moduli of hypersurfaces given that it is what determines if two hypersurfaces are isomorphic by isomorphisms of $\mathbb P^n$?