I'm reading some lecture notes on the relationship between GIT quotients and symplectic reduction, and came across the definition of a semistable point. For completeness, I will re-write it below:
Let $G$ be a complex reductive algebraic group and $(X,\mathcal{O}_X(1))$ a polarized $G$-variety. If we form the ring $$R(X) = \bigoplus_{d\geq 0} H^0(X,\mathcal{O}_X(d)),$$ the action on $X$ induces an action on $R(X)$, so we have the (sub)ring of invariants $R(X)^G$. Then a point $x\in X$ is called semistable if $s(x) \neq 0$ for some $s\in R(X)^G_{>0}$.
I frankly do not have any intuition for this definition. I found this question but the discussion there wasn't particularly helpful. Is there a Hartshorne-level way of thinking about this that lends some intuition for why this is a desirable property (or even why its called semistable)?
I found a satisfactory answer at the bottom of page 10 in the notes "Introduction to actions of algebraic groups" by Michel Brion. Found at the link here.