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The Cramer Varieties Cr(r,r+s,s)

In this paper we study quasi-homogeneous affine algebraic varieties, that is, varieties obtained as closures of orbits of suitable group representations. We also discuss one interesting case that has links with the Orthogonal Grassmannian OGr(5,10). The main aim is to write the tangent bundle and the canonical class of quasi-homogeneous affine algebraic varieties in terms of group representations.


Introduction
A homogeneous space for an algebraic group G is a space M with a transitive action of G on M . Equivalently, it is a space of the form G/H, where G is an algebraic group and H a closed subgroup of G. Homogeneous spaces play a vital role in the representation theory of the algebraic group because representations are often realised as the space of sections of vector bundles over homogeneous spaces. Homogeneous spaces have been studied in different context but not much is known about quasi-homogeneous spaces.
Reid and Corti [1] studied weighted analogs of the homogeneous spaces such as Grassmannian Gr (2,5) and the Orthogonal Grassmanian OGr(5, 10) and how to use these as weighted projective constructions.
In this paper, we study quasi-homogeneous spaces arising from two sets of matrix equations. Let M and N be r × t and t × s matrices with t = r + s, r ≤ s. We define a variety V ⊂ C (r×s)+(s×t)+1 by the two sets of equations where ω ∈ C and r M , s N denote the r × r and s × s minors of M and N respectively. We follow the method given in [2] to equate ω times the r × r minors of M with the s × s minors of N . We denote these varieties by Cr(r, r + s, s) and call them Cramer varieties. The case r = 3 and s = 1 is the original codimension 4 example in the first paper of Kustin and Miller [3]. Understanding this case led them to the more general notion of Gorenstein unprojection. For r = 4 and s = 1, these varieties in terms of equations are unprojections and have been studied by Papadakis and Reid, see [4,5]. In fact for s = 1 the variety is a single unprojection because all x i = 0 is a codimension (r + 1) complete intersection D and all m ij x j = 0 is a codimension r complete intersection X containing D. So Kustin-Miller unprojection applies to give ω as an unprojection variable with the second set of equations as unprojection equations. Also Hochster [2] studied these examples in relation to the variety of complexes.
We prove that the canonical divisor of the Cramer variety Cr(r, t, s) is Cartier in Proposition 1. Our main result is Theorem 1 in section 3 that gives criteria to calculate the canonical class of the Cr(r, t, s) in terms of weights.
2. The variety Cr(r, r + s, s) in equations Let M and N be the r × t and t × s matrices as given below: We define a variety V ⊂ C (r×t)+(t×s)+1 by the two sets of equations where ω ∈ C and r M , s N denote the r × r and s × s minors of M and N respectively. We follow the method given in [2] to equate ω times the r × r minors of M with the s × s minors of N . Let T = {1, . . . , t} be a set of t elements and T r be a subset of any r elements of T . Let M 1,...,r be the minor of M obtained from any r columns of M and let N 1,...,r be the minor of N obtained by deleting the r rows of N . We equate these minors as follows (1) either the rank of M is full and ω = 0; (2) or the rank of M is strictly less than r and ω = 0; (3) or the rank of M is strictly less than r and ω = 0.
In case 1, when M is of maximal rank then one of the minors of M is nonzero. For N of rank s − 1, we get a codimension one irreducible variety, say V 1 . A typical element of a divisor V 1 looks like In all other cases where rank N ≤ s − 1 or rank M ≤ r − 1 the codimension is greater than or equal to two so we are not worried about that part of S: these subvarieties are not divisorial so do not appear in the canonical class.
2.1. The canonical class of Cr(r, r + s, s). Suppose that the first minor M 1,··· ,r of M is nonzero. Then we can write entries of first r rows of N , n 11 ,· · · ,n 1s , n 21 , n 22 ,· · · ,n 2s , · · · n r1 , n r2 ,· · · ,n rs and ω in terms of remaining entries of M and N . Similarly if we assume that the minor M 2,··· ,r+1 of M is nonzero then we can solve for n 21 ,· · · ,n 2s , n 31 , n 32 ,· · · ,n 3s , · · · ,n r+1,1 , n r+1,2 ,· · · ,n r+1,s and ω. In the same way if we assume that any r × r minor of M is nonzero then we can use that to solve for the r rows of N and ω where the coordinates will be remaining entries of M and N .
Let U M1,··· ,r =0 and U M2,··· ,r+1 =0 be the two charts for V with coordinates ξ 1 , . . . , ξ rt , ξ rt+1 · · · , ξ rt+s 2 and η 1 , . . . , η rt , η rt+1 , η rt+s 2 . There are rt + s 2 − s coordinates common to both charts because these charts only differ by one row of N . The change of coordinates from one chart to other is given by the rt + s 2 Jacobian matrix J whose first rt + s 2 − s block is I rt+s 2 −s×rt+s 2 −s : where C is the s × s diagonal matrix whose diagonal entries are M 2,··· ,r+1 M 1,··· ,r and the The sheaf of the canonical differentials is The minor M 1,··· ,r is invertible on M 1,··· ,r and putting (M 1,··· ,r ) s in the denominator is a convenient trick to cancel out the Jacobian matrix, which will appear again later. The above calculation of the Jacobian determinant shows that σ 1,··· ,r = σ 2,··· ,r+1 and repeating the same calculation defines σ = σ i1,··· ,ir independently of i 1 , · · · , i r . Since σ i1,··· ,ir is a basis of rt+s 2 Ω 1 V and has no zeros or poles, exactly because of the M i1,··· ,ir in the denominator, we have 3. The variety Cr(r, t, s) as a quasi-homogeneous space Our aim is to study the variety V as the closure of the orbit of a special vector. Let G = GL(r) × GL(t) × GL(s) which is a reductive algebraic group. Let W r , W t and W s be the given r, t and s-dimensional representations of GL(r), GL(t) and GL(s) respectively.
We want to define an action of G = GL(r)×GL(t)×GL(s) on the representation R = Hom(W r , W t ) ⊕ Hom(W t , W s ) ⊕ C such that the variety V is invariant under this action. In coordinate-free terms, M ∈ Hom(W r , W t ), N ∈ Hom(W t , W s ) and ω ∈ C and the action of (A, B, C) ∈ G with A ∈ GL(r), B ∈ GL(t), C ∈ GL(s) is defined as follows, .
Let M and N be matrices of maximal rank. By using row and column operations we can write M and N as follows The stabiliser of v = (M 0 , N 0 , ω = 1) is where ( * ) means there is no restriction on this block. One can observe that v is not the highest weight vector, it is not even a weight vector. In this part we consider the open orbit Cramer variety Cr(r, t, s) is a quasi-homogeneous space with the natural action of G and V = G · v.
3.1. The Weyl group W (G). To study the algebraic group G = GL(r) × GL(t) × GL(s) and its representations we use the Weyl group W (G) ∼ = S r × S t × S s which acts as a permutation group. We know from section 3 that R = Hom(W r , W t ) ⊕ Hom(W t , W s ) ⊕ C is a representation of G. We describe here how the Weyl group acts on R. The group S r acts on any M ∈ Hom(W r , W t ) from the left and permutes the rows while S t acts on the right and permutes the columns. Similarly, S t acts on N ∈ Hom(W t , W s ) from the left and permutes the rows and S s acts on right and permutes the columns.
In section 2 we showed that we have only one irreducible divisor V \V 0 , say V 1 . We want to show that V 1 ⊂ G · v. For this we show that there exists a one parameter subgroup P (t) such that V 1 ⊂ P (t) · v. In fact the one parameter subgroup P (t) = (T A (t), T B (t) and T C (t)) acts on M 0 = I r×r 0 r×s , N 0 = 0 r×s I s×s and ω = 1 .
and we get  Therefore we have shown that there exists one parameter subgroup P (t) such that V 1 ⊂ P (t) · v. That is for t → 0 along the one parameter subgroup we get the typical vector of V 1 .

3.4.
The relationship between the canonical class of Cr(r, t, s) and representations of G. In this section we observe the relationship between the tangent bundle and the canonical class to the open orbit and the representations of G. We know from section 2 that the tangent bundle T V to V is a vector bundle of rank rt + s 2 and top wedge of its dual is equal to s and W t are the given r, s, s and t-dimensional GL(r)-, GL(s)-, GL(s)-and GL(t)-representations.
3.5. The canonical class in terms of weights. The canonical differential σ 1,··· ,r in equation (1) is a weight vector for the maximal T ⊂ G with weight (det(A)) r × det(B) (det(C)) s and similarly all the σ = σ i1,··· ,ir are weight vectors with weight (det(A)) r × det(B) (det(C)) s . The only problem is T ⊂ G does not normalise the stabiliser H. If we take the restricted torus acts on the canonical differential σ 1,··· ,r in equation (1) and σ 1,··· ,r is a weight vector with weight (det T A ) s (det T C ) r . Let h and g be the Lie algebras of H and G respectively. The tangent bundle T G/H to G/H comes from the representation g/h: g/h is the tangent space to G/H at the identity H, and the tangent space to any other gH ∈ G/H is given by g/ghg −1 . The canonical class of the variety G/H is G/H ). The canonical differential K G/H is a weight vector for T H and its weight is exactly the product of those weights of G which are not weights of H. Theorem 1. The weight of the canonical differential K G/H is the determinant of the restricted torus.
Proof. In this case of the Cramer variety Cr(r, t, s), there are rt + s 2 weights that are weights of G but not of H and the product of those weight is (det T A ) s (det T C ) r . In the matrix below, each asterisk block represents a collection of weight spaces of G that are not the weight spaces for H, * r×r * r×s 0 * s×s .
When we multiply these weights then the only contribution comes from the top right r × s block because the product of all the weights in the r × r and s × s square blocks is 1.
This shows that the canonical differential is a multiple of the determinant (det T A ) s (det T C ) r under the action of T H on g/h. If M is not of maximal rank then it follows from the second equation that N must be of rank less than 2. This is locus of codimension greater than or equal to 2 so we are not worried about this locus when calculating the canonical class of the variety.
We can assume the first entry m 11 of M is nonzero. We can use that to solve the top row of N and m 22 , m 23 and m 24 in terms of remaining entries of M and N . Similarly if we assume the entry m 21 of M is nonzero then we can use that to eliminate the first row of N and m 12 , m 13 and m 14 in terms of remaining entries of M and N .