Hypersurfaces of degree $ d $ in $ \mathbb{P}^n $ are parametrized by points in the space $ k[x_0 , \dots , x_n ]_d \backslash \{0\} $ of non-zéro degree $d$ homogeneous polynomials in $n+1$ variables.
As any non-zero scalar multiple of a homogeneous polynomial $F$ defines the same hypersurface, the projectivization of this space : $$ Y_{d,n} = \mathbb{P} \big( k[x_0 , \dots , x_n]_d \big) $$ is a smaller dimensional parameter space for these hypersurfaces.
Please :
How to show that $ Y_{d,n} $ parametrises a tautological family of degee $d$ hypersurfaces in $ \mathbb{P}^n $ with the local universal property ?.
How to deduce then that any coarse moduli space for hypersurfaces is a categorical quotient by an action of $ \mathrm{SL}_n $ on $ Y_{d,n} $ to specify ?.
Thanks in advance for your help.