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.

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.

For details you may look at:

• S. Mukai, Finite generation of the Nagata invariant rings in A-D-E cases, RIMS Preprint n. 1502, Kyoto, 2005.
• C. Casagrande, Rank 2 quasiparabolic vector bundles on P1 and the variety of linear subspaces contained in two odd-dimensional quadrics, Mathematische Zeitschrift volume 280, 981–988 (2015).
• C. Araujo and C. Casagrande, On the Fano variety of linear spaces contained in two odd-dimensional quadrics, Geometry & Topology 21 (2017) 3009–3045.