The classical BB-decomposition works for non-singular projective varieties. Here I want to consider projective schemes, in particular when the scheme is not reduced.
Let $\Bbbk=\mathbb{C}$. Let $X$ be a projective $\Bbbk$-scheme. Let $\mathbb{G}_m$ act on $X$. Consider the following functor $$F:(\mathit{Sch}/X)^{opp}\to \mathit{Sets},\quad (T\to X)\mapsto \Big\{\mathbb{A}^1\times_\Bbbk T\to X:\begin{matrix}\textrm{the morphism is }\mathbb{G}_m\textrm{-equivariant}\\\{1\}\times T\to X\textrm{ is the structure morphism}\end{matrix}\Big\}$$ where $\mathbb{G}_m\curvearrowright\mathbb{A}^1\times T$ is the scalar multiplication $\mathbb{G}_m\curvearrowright\mathbb{A}^1$ and the trivial $\mathbb{G}_m\curvearrowright T$.
The question is the following.
Is there a stratification of locally closed subschemes $\coprod X_i\to X$ representing the functor $F$?
If the answer is affirmative, I hope the following can also be touched:
- Is there an open stratum $X_0\subseteq X$? How can we construct it?
- If $X_0$ exists, is there a natural closed subscheme structure on $X\setminus X_0$? Can we write down the sheaf of ideals or can we describe its functor of points (as a closed subfunctor of $F$)?
I have made the following attempt.
Assume further the existence of an ample line bundle $\mathcal{L}$ on $X$ with a $\mathbb{G}_m$-linearization. Then $X\hookrightarrow\mathrm{Proj}\Big(\bigoplus_{n\geq0}H^0(X,\mathcal{L}^n)\Big)$ is $\mathbb{G}_m$-equivariant. Let $H^0(X,\mathcal{L}^n)_{\max}\ne0$ denote the weight space of maximal weight on $H^0(X,\mathcal{L}^n)$.
If for $n\gg0$, we have $H^0(X,\mathcal{L}^n)_{\max}$ is not nilpotent, then $X_0$ should be the non-vanishing locus of $H^0(X,\mathcal{L}^n)_{\max}$, and the choice of $n\gg0$ does not change $X_0$.
Else $H^0(X,L^n)_{\max}$ is nilpotent for all $n>0$, then I guess $X_0=\emptyset$.
I have the following questions about this method:
- Is $\mathcal{L}$ necessary? I can not define $X_0$ without this linearization. However, there is no such thing in the functor $F$ and the classical BB-decomposition.
- The closed subset $X\setminus X_0$ is cut out by $H^0(X,\mathcal{L}^n)_{\max}$ for some $n\gg0$. This gives a closed subscheme structure. However, this subscheme depends on the choice of $n\gg0$: the larger the $n$, the "fatter" the closed subscheme $X\setminus X_0$.
Any comments or references are welcome.
