The quotient stack $[\mathbb{A}^n / \mathrm{GL}_n]$

Question

Consider the affine space $\mathbb{A}^n$ (over some base scheme) with the usual $\mathrm{GL}_n$-action. What does the quotient stack $[\mathbb{A}^n / \mathrm{GL}_n]$ classify? If $n=1$, then we get $[\mathbb{A}^1 / \mathbb{G}_m]$, which classifies line bundles together with a global section, right? In general, $[\mathbb{A}^n / \mathrm{GL}_n]$ classifies vector bundles of rank $n$ together with some additional data - is it again just a global section?

I would also be happy if someone can add a geometric picture of $[\mathbb{A}^n / \mathrm{GL}_n]$, at least over some algebraically closed field. What are the points of this stack, and what are their stabilizers? What's your geometric intuition for $[\mathbb{A}^n / \mathrm{GL}_n]$?

PS: I am quite new to stack-land and hope that this question is not too trivial.

PPS: In the comments it is suggested to do the same as for $n=1$, so let me better explain what I've done for $n=1$. A $T$-point of $[\mathbb{A}^1/\mathbb{G}_m]$ is a $\mathbb{G}_m$-bundle $P \to T$ with a $\mathbb{G}_m$-map $P \to \mathbb{A}^1$. Then $P$ corresponds to an invertible sheaf $\mathcal{L}$ on $T$ via $P = \mathrm{Spec}_T(\bigoplus_{z \in \mathbb{Z}} \mathcal{L}^{\otimes z})$. Assuming that $T$ is affine, the $\mathbb{G}_m$-map therefore corresponds to a section of $\bigoplus_{z \in \mathbb{Z}} \mathcal{L}^{\otimes z}$, say $\sum_z a_z$, which is compatible with the $\mathcal{O}(\mathbb{G}_m)$-coaction, which comes down to $\sum_z a_z u^z = \sum_z a_z u$, i.e. $a_z=0$ for $z \neq 1$. We end up with a section $a_1$ of $\mathcal{L}$. For $n >1$, I have tried the same, but the quasi-coherent algebra induced by some locally free sheaf of rank $n$, whose spectrum is the corresponding $\mathrm{GL}_n$-bundle, is quite complicated, at least globally. Locally, we just have $P = \mathrm{GL}_n \times T$, and the $\mathrm{GL}_n$-map $P \to \mathbb{A}^n$ corresponds to a map $T \to \mathbb{A}^n$, i.e. $n$ global sections of $T$, which is one section of the free sheaf $\mathcal{O}_T^n$ of rank $n$. But a) I am not sure if this glues properly, and b) I would prefer a global argument as for $n=1$.

Answer 1

The category of maps from a test object $T$ to a quotient stack $[X/G]$ has the following general form. Objects are pairs $(P, f)$, where $P$ is a $G$-torsor over $T$, and $f: P \to X$ is a $G$-equivariant map. Morphisms $(P,f) \to (P',f')$ are torsor isomorphisms $g: P \to P'$ satisfying $f = f' g$. Here, $X$ is the vector representation $\mathbf{O}^n$, and $G = GL_n$.

However, the vector representation of $GL_n$ is faithful, so you may replace the $GL_n$-torsor with the associated bundle $P \times^{GL_n} \mathbf{O}^n$ to get an equivalent stack. Here, objects are pairs $(V, f)$, where $V$ is a rank $n$ vector bundle over $T$, and $f: V \to X$ is a $GL_n$-equivariant map to the vector representation, or more usefully, an equivariant sheaf map to the trivial rank $n$ bundle. Morphisms $(V,f) \to (V',f')$ are given by vector bundle isomorphisms $g$ satisfying $f = f' g$.

By taking dual vector bundles, we get an equivalent stack, whose objects are pairs $(V,h)$, where $V$ is a vector bundle, and $h$ is a $GL_n$-equivariant map from the trivial rank $n$ bundle to $V$. Since $GL_n$ acts transitively on the nonzero vectors in the vector representation (and its dual), any map $h$ is uniquely determined by the image of a fixed nonzero vector in the dual vector representation, i.e., a distinguished section of the vector bundle. Then we have a stack whose objects are vector bundles with sections, and morphisms are vector bundle isomorphisms that take sections to sections.

Edit: Here is a picture of the stack: $- \cdot$

The dash is an open dense copy of $B(Aff_{n-1})$ (since the stabilizer of a nonzero vector is the affine group $Aff_{n-1}$) and the dot is a closed copy of $BGL_n$.

Answer 2

I'll use the definition of stack as a (weak) functor from the category of schemes to that of groupoids (as opposed to the definition as a fibered category over the category of schemes).

The prestack associated to the action of $GL_n$ on $\mathbb A_n$ is, by definition, given by $$ X \mapsto \left\{\begin{matrix}\text{Objects: maps $s:X \to \mathbb A^n$}\\ Hom(s_1,s_2): \text{maps $f:X\to GL_n$ such that $f\cdot s_1 = s_2$} \end{matrix}\right\} $$ That prestack is easily seen to be equivalent to $$ X \mapsto \left\{\begin{matrix}\text{Objects: Sections $s:X\to \mathbb A^n\times X$ of the trivial vector bundle $\mathbb A^n\times X\to X$}\\ \text{$Hom(s_1,s_2)$: Vector bundle isos $f:\mathbb A^n\times X\to \mathbb A^n\times X$ s.t. $s_2=f\circ s_1$}\end{matrix}\right\} $$ The associated stack is then given by $$ X \mapsto \left\{\begin{matrix}\text{Objects: Vector bundles $V \to X$, together with a section $s:X\to V$}\\ \text{$Hom((V_1,s_1),(V_2,s_2))$: Vector bundle iso $f:V_1 \to V_2$ such that $s_2=f\circ s_1$}\end{matrix}\right\} $$

Indeed, any vector bundle with section is locally of the form trivial vector bundle with section. Moreover, any vector bundle with section $(V,s)$ over $X$ can be described by an open cover of $X$, on each open of the cover the data of a trivial vector bundle with section, and gluing isomorphisms subject to the obvious cocycle condition. That's exactly what does the associated stack to the prestack of tivial vector bundles with section.

Answer 3

Let $\text{G}$ be a group. Then maps $$S\ \longrightarrow \ \text{BG}$$ from a test scheme $S$ is the same thing as a $\text{G}$-bundle $P\to S$ (or rather the groupoid of such).

Now let $X$ be another scheme with an action of $\text{G}$. Then to any $\text{G}$ bundle we can build its associated $X$ bundle: $$(P\to S)\ \rightsquigarrow \ (P\times_\text{G}X\to S).$$ This $X$ bundle has another description, as the pullback $\require{AMScd}$ \begin{CD} P\times_\text{G}X @>>> X/\text{G}\\ @V V V @VV V\\ S @> P> > \text{BG} \end{CD} Answer: Choosing a lift of the map $P$ to $$S\ \longrightarrow \ X/\text{G}$$ is thus the same as choosing a section of the $X$-bundle $P\times_\text{G}X$ associated to $P$.

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Examples:

1. $\mathbf{A}^1/\mathbf{G}_m$ classifies $\mathbf{G}_m$ bundles (a.k.a. line bundles) along with a section of the associated line bundle.
2. Similarly, $\mathbf{G}_m/\mathbf{G}_m$ classifies $\mathbf{G}_m$ bundles with a nonvanishing section. This trivialises the line bundle, which geometrically corresponds to the fact that $\mathbf{G}_m/\mathbf{G}_m=\text{pt}$ is a single point.
3. $\mathbf{A}^n/\text{GL}_n$ classifies $\text{GL}_n$ bundles with a section of its associated vector bundle.
4. $\mathbf{P}^1=(\mathbf{A}^2\setminus 0)/\mathbf{G}_m$ classifies line bundles together with two sections which do not vanish simultaneously.
5. Similarly for $\mathbf{P}^n$.
6. A variation on 1. If instead $\mathbf{G}_m$ acts on $\mathbf{A}^1$ with weight $n$, then $\mathbf{A}^1/\mathbf{G}_m$ classifies line bundles with a section of its $n$th tensor power. Similarly for e.g. weighted projective spaces.
7. More generally, if $\text{P}$ is a parabolic subgroup of the reductive group $\text{G}$, then the generalised flag variety $\text{G}/\text{P}$ classifies $\text{P}$ bundles with a trivialisation of the associated $\text{G}$ bundle. This gives the functor of points for e.g. Grassmannians $\text{Gr}(k,n)$.