# Quiver Grassmannians can be anything

A standard Grassmannian $Gr(m,V)$ is the manifold having as its points all possible $m$-dimensional subspaces of a given vectorspace $V$. As an example, $Gr(1,V)$ is the set of lines through the origin in $V$ and therefore is the projective space $\mathbb{P}(V)$. Grassmannians are among the nicest projective varieties, they are smooth and allow a cell decomposition.

A quiver $Q$ is just an oriented graph. Here's an example

A representation $V$ of a quiver assigns a vector-space to each vertex and a linear map between these vertex-spaces to every arrow. As an example, a representation $V$ of the quiver $Q$ consists of a triple of vector-spaces $(V_1,V_2,V_3)$ together with linear maps $f_a~:~V_2 \rightarrow V_1$ and $f_b,f_c~:~V_2 \rightarrow V_3$.

A sub-representation $W \subset V$ consists of subspaces of the vertex-spaces of $V$ and linear maps between them compatible with the maps of $V$. The dimension-vector of $W$ is the vector with components the dimensions of the vertex-spaces of $W$.

This means in the example that we require $f_a(W_2) \subset W_1$ and $f_b(W_2)$ and $f_c(W_2)$ to be subspaces of $W_3$. If the dimension of $W_i$ is $m_i$ then $m=(m_1,m_2,m_3)$ is the dimension vector of $W$.

The quiver-analogon of the Grassmannian $Gr(m,V)$ is the Quiver Grassmannian $QGr(m,V)$ where $V$ is a quiver-representation and $QGr(m,V)$ is the collection of all possible sub-representations $W \subset V$ with fixed dimension-vector $m$. One might expect these quiver Grassmannians to be rather nice projective varieties.

However, last week Markus Reineke posted a 2-page note on the arXiv proving that every projective variety is a quiver Grassmannian.

Let's illustrate the argument by finding a quiver Grassmannian $QGr(m,V)$ isomorphic to the elliptic curve in $\mathbb{P}^2$ with homogeneous equation $Y^2Z=X^3+Z^3$.

Consider the Veronese embedding $\mathbb{P}^2 \hookrightarrow \mathbb{P}^9$ obtained by sending a point $(x:y:z)$ to the point

$(x^3:x^2y:x^2z:xy^2:xyz:xz^2:y^3:y^2z:yz^2:z^3)$

The upshot being that the elliptic curve is now realized as the intersection of the image of $\mathbb{P}^2$ with the hyper-plane $\mathbb{V}(X_0-X_7+X_9)$ in the standard projective coordinates $(x_0:x_1:\cdots:x_9)$ for $\mathbb{P}^9$.

To describe the equations of the image of $\mathbb{P}^2$ in $\mathbb{P}^9$ consider the $6 \times 3$ matrix with the rows corresponding to $(x^2,xy,xz,y^2,yz,z^2)$ and the columns to $(x,y,z)$ and the entries being the multiplications, that is

$$\begin{bmatrix} x^3 & x^2y & x^2z \\ x^2y & xy^2 & xyz \\ x^2z & xyz & xz^2 \\ xy^2 & y^3 & y^2z \\ xyz & y^2z & yz^2 \\ xz^2 & yz^2 & z^3 \end{bmatrix} = \begin{bmatrix} x_0 & x_1 & x_2 \\ x_1 & x_3 & x_4 \\ x_2 & x_4 & x_5 \\ x_3 & x_6 & x_7 \\ x_4 & x_7 & x_8 \\ x_5 & x_8 & x_9 \end{bmatrix}$$

But then, a point $(x_0:x_1: \cdots : x_9)$ belongs to the image of $\mathbb{P}^2$ if (and only if) the matrix on the right-hand side has rank $1$ (that is, all its $2 \times 2$ minors vanish). Next, consider the quiver

and consider the representation $V=(V_1,V_2,V_3)$ with vertex-spaces $V_1=\mathbb{C}$, $V_2 = \mathbb{C}^{10}$ and $V_2 = \mathbb{C}^6$. The linear maps $x,y$ and $z$ correspond to the columns of the matrix above, that is

$$(x_0,x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8,x_9) \begin{cases} \rightarrow^x~(x_0,x_1,x_2,x_3,x_4,x_5) \\ \rightarrow^y~(x_1,x_3,x_4,x_6,x_7,x_8) \\ \rightarrow^z~(x_2,x_4,x_5,x_7,x_8,x_9) \end{cases}$$

The linear map $h~:~\mathbb{C}^{10} \rightarrow \mathbb{C}$ encodes the equation of the hyper-plane, that is $h=x_0-x_7+x_9$.

Now consider the quiver Grassmannian $QGr(m,V)$ for the dimension vector $m=(0,1,1)$. A base-vector $p=(x_0,\cdots,x_9)$ of $W_2 = \mathbb{C}p$ of a subrepresentation $W=(0,W_2,W_3) \subset V$ must be such that $h(x)=0$, that is, $p$ determines a point of the hyper-plane.

Likewise the vectors $x(p),y(p)$ and $z(p)$ must all lie in the one-dimensional space $W_3 = \mathbb{C}$, that is, the right-hand side matrix above must have rank one and hence $p$ is a point in the image of $\mathbb{P}^2$ under the Veronese.

That is, $Gr(m,V)$ is isomorphic to the intersection of this image with the hyper-plane and hence is isomorphic to the elliptic curve.

The general case is similar as one can view any projective subvariety $X \hookrightarrow \mathbb{P}^n$ as isomorphic to the intersection of the image of a specific $d$-uple Veronese embedding $\mathbb{P}^n \hookrightarrow \mathbb{P}^N$ with a number of hyper-planes in $\mathbb{P}^N$.

ADDED (May 23rd 2013) : In migrating this blog from WordPress to Drupal I forgot to include the comments to this post. As they are mentioned in at least one paper, that by Claus Ringel Quiver Grassmannians and Auslander varieties for wild algebras I have now manually added the more important comments from a database backup. I've included the original times when these comments were posted in the comment. My apologies to the commenters.

### Michel Van den Bergh

Here is a slightly different proof I think (once you know something is true it becomes easier to prove).

(0) I will use the “quotient Grassmannian”. This is of course equivalent.

(1) It is sufficient to construct a quiver Grassmannian for a quiver with relations with the desired property. Indeed if a representation satisfies certain relations then so does any sub or quotient representation.

(2) Assume we want to do P^n (projective space). Then we take the Beilinson quiver with 3 vertices and 2 x n+1 arrows and we impose the commutativity relations.

The moduli space of representations of dimension vector (1,1,1) generated in the first vertex is P^n.

This moduli space is also the quiver Grassmannian of quotients with dimension vector (1,1,1) of the projective representation corresponding to the first vertex.

(3) For an arbitrary projective variety we use the fact that it can be defined by quadratic relations in some P^n and impose these relations on the corresponding Beilinson quiver from (2) (which already has the commutativity relations). Then we use again the quiver Grassmannian of quotients with dimension vector (1,1,1) of the projective representation corresponding to the first vertex.

(4) If we do not want to use the fact that we can use quadratic relations then we can use larger Beilinson quivers.

### Lutz Hille

Examples like the one Michel just mentioned can be found in two 'old' papers, one by myself (a projective example: Tilting line bundles ...) and one by Birge Huisgen-Zimmermann (affine examples: Varieties of uniserial modules ...).

### Jason Starr

ORIGINALLY POSTED BY Jason Starr ON 2012-05-02, 14:20:25

Do you know if the proof is related to the proof of Mnev's Universality Theorem? Mnev's paper is not cited in the bibliography, so perhaps not.

### Javier

ORIGINALLY POSTED BY Javier ON 2012-05-04, 09:10.24

That is a cool result! Can it be strengthened to include morphisms, so that every morphism of projective varieties comes from a morphism of quiver representations?

### David Roberts

ORIGINALLY POSTED BY David Roberts ON 2012-05-07, 11:32.57

@javier - I was wondering about this at the n-category cafe. I think the assignment {quiver rep for Q} --> {proj. variety} is functorial (I'm not an algebraic geometer!) and the question is what is this putative functor like? By the above result it is essentially surjective.

One thing that would need looking at is what happens when we change quivers as well as representations - what are the maps between quivers you would like to allow? There is a natural notion of morphism between quiver reps where we don't stick with the same quiver: consider a quiver as a free category on the directed graph, and a representation as a functor to Vect. Then we can consider the slice (2-)category over Vect where an arrow is a triangle commuting up to a 2-arrow.

These may give maps of varieties, but what sort I do not know.

Once we have this, then we can ask again what properties this functor has (again, it's essentially surjective). I would love it if it had some sort of adjoint.

Then one can start to transport cohomological invariants back and forth, and let the fun begin.

### Markus Reineke

ORIGINALLY POSTED BY Markus Reineke ON 2012-05-07, 11:32.57

check your blog for some time now, big surprise when I came here again. Thx for the blog-coverage!

@Michel: I think our constructions are (almost) compatible: to realize projective space, you consider the Beilison quiver 1=>2=>3 and the projective representation P_1 when commutativity relations are imposed, I consider 1< = 2 = >3 and the special representation V_d (for the d-th Veronese). Restricting to the subquiver supported on 2 and 3, I think your P_1 is isomorphic to my V_2 (quadratic Veronese). In my construction there's one more option, namely instead of using the defining homogeneous equations of the variety as representing the arrows 1 < = 2, we can realize the projective variety as the image of the Veronese intersected with a linear subspace, spanned by some v_i, which leads to the quiver Grassmannian for 1 = >2 = >3 (and e=(1,1,1)), where the v_i represent the arrows from 1 to 2. That should be exactly the representation of the Beilinson quiver you proposed.

If only I'd been clever enough to realize an arbitrary projective variety by quadratic equations....

@Javier: good question... in fact it was our discussions with Oliver in Wuppertal that made me think about this universality thing. Does it have any implication for F_1-related things that my representation V is defined over F_1?

I'm a bit surprised about the discussion at the n-category cafe - I think my remark on quiver Grassmannians is just the 42nd+ incarnation of "with wild quivers, everything goes wrong". I thought about this because Bernhard Keller asked me some years ago, then some discussions with Javier and Oliver Lorscheid made me think of it again, a boring conference talk a few weeks ago gave the right idea.

I also wanted to have a mathematical justification for considering what looks at first sight like a ridiculously special case, namely the quiver Grassmannians Gr_(dim P)(P+I) of subrepresentations of the sum of a projective and an injective representation of a Dynkin quiver of arXiv:1106.2399 - now that we know that there can't be a general theory of all quiver Grassmannians, it's clear that one has to restrict to special cases.