# The Cox ring of the space of

complete rank two collineations

###### 1991 Mathematics Subject Classification:

14C20## 1. Introduction

The space of complete collineations compactifies the space of isomorphisms between -dimensional vector subspaces of two given complex vector spaces and of finite dimension and respectively; it showed up in classical algebraic geometry, see [7, 8] for some background. In [9], one finds the following precise definition: let denote the tautological bundle over the Grassmannian , consider the rational map

and define to be the closure of the image. Various descriptions of this space have been given by Vainsencher [10] and Thaddeus [9]. Among other things it was shown in [9] that can be realized as the Chow/Hilbert quotient and also as the GIT-limit of a torus action on a Grassmannian; this establishes a nice analogy to results of Kapranov [6] on the space of point configurations on the line.

In this note, we determine the Cox ring of the space of complete rank 2 collineations. The Cox ring of a complete normal variety with free finitely generated divisor class group is defined as follows: take any subgroup of the group of Weil divisors projecting isomorphically onto and set

The interesting cases for the space of complete rank 2 collineations are ; the remaining ones are simple, see Remark 3. Here comes our result.

###### Theorem 1.

Let . The Cox ring of the space of complete rank 2 collineations is the factor algebra with the ideal generated by

Moreover, the grading of the Cox ring by the divisor class group is given by

This shows in particular that has a finitely generated Cox ring, i.e. it is a Mori dream space. Moreover, the explicit knowledge of the Cox ring in terms of generators and relations opens an access to the geometry; for example [4, Proposition 4.1] gives the following.

###### Corollary 1.

Let . Set . Then the cones of effective, movable and semiample divisor classes of in are

## 2. Proof of Theorem 1

By the main theorem of [9], the space of complete rank collineations is obtained by blowing up a suitable GIT quotient of the Grassmannian by the -action with the weights one on and minus one on . Our idea is to realize this blow up as a neat controlled toric ambient modification in the sense of [4, Def. 5.4, 5.8] and then verify that we are in the situation of [4, Thm. 5.9] which finally gives the Cox ring.

We work with the affine cone over . It lies in the Plücker coordinate space and is defined by the Plücker relations

Note that is invariant under the action of the 2-torus on the Plücker space given by the following weight matrix , where and and :

According to [2, Prop. 2.9], the GIT-quotients of the -action correspond to the cones of the GIT-fan in the rational character space .

###### Proposition 1.

The -actions on and have the same GIT-fan ; it has the maximal cones and .

###### Proof.

In general, the GIT-fan of a -action on an affine variety consists of the GIT-chambers , where . Each such GIT-chamber is defined as

From this we directly see that is the GIT-fan of the -action on . To obtain this as well for the -action on , it is sufficient to show, that are of the form with suitable . For take the point with the Plücker coordinates and else; for take with and else. ∎

We denote by and the GIT quotients corresponding to the chamber . With any taken from the relative interior of , we realize and as sets of semistable points

Note that , are toric varieties and is a toric morphism. Moreover, we have , the quotient map is the restriction of (which justifies denoting both of them by ) and the induced maps are closed embeddings.

###### Remark 1.

Let . Then the quotients and are characteristic spaces, i.e. relative spectra of Cox sheaves, see [1]. Moreover, the embedding is neat in the sense that the toric prime divisors of cut down to prime divisors on and the pullback on the level of divisor class groups is an isomorphism, see [4, Def. 2.5, Prop. 3.14]. For , one has analogous statements on and .

###### Remark 2.

The fan of is obtained from via linear Gale duality, see for example [1, Sec II.2]. The rays of are generated by the columns of any matrix having the rows of as a basis of its nullspace; we take

Each maximal cone of is obtained by removing one column from inside the -block, one from outside the -block and then taking the cone over the remaining ones. Similarly, for the maximal cones of , remove one column from inside the -block and one from outside.

We perform a toric blow up of . Consider the cone generated by the last colums of and the barycentric subdivision of at ; that means that we insert the ray through . Then defines a toric blow up centered at the toric orbit closure corresponding to . Let be the proper transform of under this map.

###### Proposition 2.

The variety is the space of rank-2 complete collineations.

###### Proof.

The closed subset of is the toric orbit closure corresponding to the cone in generated by the last coordinate axes. Note that we have . Consequently, we obtain

Hence is the blow up of at . From [9, Sec. 3.3] we know, that the space of complete collineations is obtained from by blowing up the subvariety . Since is a good quotient and , are closed invariant subsets, we obtain

∎

###### Proposition 3.

The varieties and the center of are smooth.

###### Proof.

This is a result of Vainsencher, see [10, Theorem 1]. We give a simple alternative proof. By Remark 2, the set of semistable points is covered by open affine -saturated sets as follows

By the nature of the Plücker relations, is smooth and it is contained in the first union. There the torus acts freely and thus the quotient inherits smoothness. The inverse image of the center of is explitly given by cutting down the Plücker relations. Calculating the Jacobian of the resulting equations shows that is smooth. It follows that is smooth. ∎

###### Remark 3.

For we obtain and for we obtain . In both cases Remark 1 tells us that the Cox ring is modulo the Plücker relations endowed with the -grading given by the matrix . Finally, the Cox ring of is with the standard -grading.

From now on we assume . Consider the toric characteristic space obtained via Cox’s construction, see [3] and [1, Sec. II.1.3]. Then is an open toric subvariety of . We denote the additional coordinate by ; it corresponds to the new ray . Moreover, is a good quotient for the -action on given by the weight matrix

Now set and write for the closure of in . Then we arrive at the following commutative diagram: