# In GIT, why are the semistable/unstable loci defined pointwise, instead of defining semistable/unstable subschemes?

Let $$k$$ be an algebraic closed charateristic zero field. Let $$G$$ be a reductive group over $$k$$ and let $$X$$ be a scheme (not necessarily separated) of finite type over $$k$$. Let $$L$$ be a line bundle over $$X$$. Let $$G$$ act on $$X$$ and linearise $$L$$, denoted by $$G\curvearrowright(X,L)$$.

In Mumford's Geometric invariant theory and Dogachev's Lectures on invariant theory (chapter 8), semistability is defined for geometric points, and they form an open subset $$X^{\mathrm{ss}}$$. The same is true for stable points $$X^{\mathrm{s}}$$. Unstable points are defined as the complement of semistable points, $$X^{\mathrm{us}}:=X\setminus X^{\mathrm{ss}}$$. I wonder why these notions are defined pointwise in stead of define “(semi)stable open subscheme” and “unstable subscheme”.

One reason for $$X^{\mathrm{(s)s}}$$ I guess is because open subschemes behave nicely. If we write $$X_0\subset X$$ for the subspace of closed points of $$X$$, then since $$X$$ is Jacobson, the set of open subsets of $$X$$ and the set of open subsets of $$X_0$$ are in bijection \begin{align*} \{\textrm{open subsets of } X\}&{}\overset{\textrm{1-1}}{\longleftrightarrow}\{\textrm{open subsets of }X_0\}\\O&{}\mapsto O\cap X_0. \end{align*} Also, open subsets of $$X$$ have natural open subscheme structures. So defining the “semistable open subscheme” of $$X$$ is equivalent to defining semistable closed points of $$X$$.

My question arises when I want to consider the unstable locus $$X^{\mathrm{us}}$$, and the strictly semistable locus $$X^{\mathrm{sss}}:=X^{\mathrm{ss}}\setminus X^{\mathrm{s}}$$. They are now defined as topological subspaces, $$X^{\mathrm{us}}\subset X$$ is closed and $$X^{\mathrm{sss}}\subset X$$ is locally closed. I think it is natural to consider subscheme structures on $$X^{\mathrm{us}}$$ and $$X^{\mathrm{sss}}$$.

There is research about $$X^{\mathrm{us}}$$ and $$X^{\mathrm{sss}}$$ when $$X$$ is a smooth variety. For example the instability decomposition (Kempf), characterizing how unstable a point is, and the partial desingulariastion (Kirwan), a $$G$$-equivariant blowing up $$\tilde X\to X$$ with a linearisation $$\tilde L$$ on $$\tilde X$$ such that $$\tilde X^{\mathrm{ss}}=\tilde X^{\mathrm{s}}$$. To generalise their results to appropriate schemes (e.g. non-reduced varieties), I think it is necessarily to consider subschemes on $$X^{\mathrm{us}}$$ and $$X^{\mathrm{sss}}$$.

Kempf, George R., Instability in invariant theory, Ann. Math. (2) 108, 299-316 (1978). ZBL0406.14031.

Kirwan, Frances Clare, Partial desingularisations of quotients of nonsingular varieties and their Betti numbers, Ann. Math. (2) 122, 41-85 (1985). ZBL0592.14011.

Mumford, D.; Fogarty, J.; Kirwan, F., Geometric invariant theory., Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge. 34. Berlin: Springer-Verlag. 320 p. (1994). ZBL0797.14004.

Dolgachev, Igor, Lectures on invariant theory, London Mathematical Society Lecture Note Series. 296. Cambridge: Cambridge University Press. xvi, 220 p. (2003). ZBL1023.13006.

• The semistable locus is an open subscheme (possibly empty), so it is equivalent to give the underlying open subset, and this is uniquely determined by the geometric points in a base change over an algebraically closed field. Apr 10, 2022 at 0:51
• The unstable locus does have a natural scheme structure, and this does sometimes come up, e.g., the nilpotent cone of a semisimple Lie algebra (which is a complete intersection scheme). Apr 10, 2022 at 0:54