The space you are interested in, call it $M_{(\frac{1}{2},\dots,\frac{1}{2})}$, can be described as a small transformation of the blow-up $X^{n-3}_n$ of $\mathbb{P}^{n-3}$ at $n$ points in linear general position. 

More specifically, $X^{n-3}_n$ is a Mori dream space. So its movable cone has a decomposition in convex chambers which are the nef cones of all the possibile small transfomations of $X^{n-3}_n$. From the GIT poit of view going from one chamber to the other corresponds to changing the stability conditions that is changing the weights. 

The space $M_{(\frac{1}{2},\dots,\frac{1}{2})}$ is a Fano variety of dimension $n−3$, which is smooth if $n$ is odd, and has isolated singularities if $n$ is even. From the point of view of the above decomposition its nef cone (when $n$ is odd) is the chamber containing the anticanonical divisor of $X^{n-3}_n$. 

For $n$ even the anticanonical divisor of $X^{n-3}_n$ lies in the intersection of several walls. From the birational point of view this means that $M_{(\frac{1}{2},\dots,\frac{1}{2})}$ is not $\mathbb{Q}$-factorial. From the GIT point of view it means that there are strictly semistable points.