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](https://mathoverflow.net/questions/27531), 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 $n=1$.



The quotient stack $[\mathbb{A}^n / \mathrm{GL}_n]$
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](http://en.wikipedia.org/wiki/Affine_group) $Aff\_{n-1}$) and the dot is a closed copy of $BGL\_n$.


