# 2024-07-26-11-04-29

:::{.remark}
Let $X$ be Fano and let $D \sim_\QQ r K_X$ for $r\in \QQ_{>0}$, for example a degree $d$ hypersurface.
If $K_X + D$ is ample, we can form a KSBA moduli space and we consider the "main" component containing a fixed point.
If $-(K_X+D)$ is ample, we can instead form a K-moduli space.
:::

:::{.theorem}
If $X = \PP^n$ and $D$ is a degree $d$ hypersurface, for $\eps \ll 1$, K-stability of $(\PP^n, \eps D)$ is equivalent to GIT stability of $D$.
In fact, the K-moduli space is isomorphic to the GIT stack of degree $d$ hypersurfaces.
:::

:::{.remark}
Let $(\PP^n, cD)$ both be smooth.
Then

1. If $0 < c < \min\ts{ {n+1\over d}, 1}$, then the pair is K-stable.
2. If ${n+1\over d} < c < 1$, then the pair is KSBA stable.
3. $D$ is GIT stable.
:::