Equations and syzygies of some Kalman varieties

Given a subspace L of a vector space V, the Kalman variety consists of all matrices of V that have a nonzero eigenvector in L. Ottaviani and Sturmfels described minimal equations in the case that dim L = 2 and conjectured minimal equations for dim L = 3. We prove their conjecture and describe the minimal free resolution in the case that dim L = 2, as well as some related results. The main tool is an exact sequence which involves the coordinate rings of these Kalman varieties and the normalizations of some related varieties. We conjecture that this exact sequence exists for all values of dim L.


Introduction.
Let V be a vector space over a field of arbitrary characteristic. For a subspace L V , the associated Kalman variety consists of all matrices that have a nonzero eigenvector in L. A more general definition and basic properties of Kalman varieties are contained in Section 1.1. The algebraic and geometric properties of this variety were studied by Ottaviani and Sturmfels in [OS], and their definition was motivated by Kalman's observability condition in control theory [Kal].
In particular, Ottaviani and Sturmfels find minimal generators for the prime ideal of the Kalman variety when dim L = 2 and conjecture the number of equations needed when dim L = 3. (When dim L = 1, the Kalman variety is an affine space.) Our main results involve calculating the minimal free resolution in the case dim L = 2 (Theorem 3.3) and proving their conjecture in the case dim L = 3 (Theorem 3.6). We point out that even though the Kalman varieties are of determinantal type in these cases, they are not Cohen-Macaulay varieties when dim V − 1 > dim L > 1, so the resolution is not obtained from the Eagon-Northcott complex.
The main tool is the geometric approach to free resolutions via sheaf cohomology (Section 1.3). However, it is not a straightforward application because this approach only provides information for the normalization of the Kalman variety, and the Kalman variety is not normal whenever dim L > 1. The main insight into this problem is that the Kalman varieties and their higher analogues (defined in Section 1.1) appear to have a certain inductive structure. We prove that this structure exists when dim L ≤ 3 (Theorem 3.2) and conjecture that it exists in general (see Conjecture 3.1). As further evidence, we sketch a proof of this conjecture in the case when dim V = dim L + 1 and the ground field is of characteristic 0 (see Section 3.4).
This inductive structure should provide a means to study the equations of the Kalman variety when dim L > 3. The validity of Conjecture 3.1 would make the Kalman varieties a good testing ground for studying the equations and free resolutions of non-normal varieties. In particular, there are very few known instances where the approach described in Section 1.3 works effectively for non-normal varieties. One particularly important instance where the approach in Section 1.3 is relevant but where the varieties can fail to be normal are the nilpotent orbits in Lie theory [Wey, Chapter 8], so hopefully the insights gained from studying the easier case of Kalman varieties will be useful in more complicated situations.
The outline of the article is as follows. In Section 1, we summarize the properties of Kalman varieties that we will be using, as well as the necessary constructions and theorems needed to use the geometric approach to free resolutions. In Section 2, we prove a few preparatory results on the normalizations of Kalman varieties, which we use in Section 3 to prove our main results.

Acknowledgements.
The author thanks Bernd Sturmfels for showing him [OS,Conjecture 3.6] which was the motivation for this article. The author also thanks Giorgio Ottaviani and Bernd Sturmfels for helpful comments on an earlier draft. The author was supported by an NSF graduate research fellowship and an NDSEG fellowship while this work was done.

Kalman varieties.
Fix a field K, a vector space V , and a subspace L V . Set W = (V /L) * , and let End(V ) be the space of linear operators on V with coordinate ring A = Sym(End(V ) * ), which is graded via deg End(V ) * = 1. Also, let n = dim V , d = dim L and pick 1 ≤ s ≤ d. The Kalman variety is Equations that define K s,d,n (at least set-theoretically) can be obtained as follows. Pick an ordered basis for V starting with a basis for L followed by a basis for V /L and write ϕ in block matrix form α β γ δ . Then K s,d,n is the zero locus of the (d − s + 1) × (d − s + 1) minors of the reduced The bundle S is not completely reducible, but it has a filtration whose associated graded is For later use, set ξ = ((End(V ) × Gr(s, L))/S) * . Then Let p 1 : End(V ) × Gr(s, L) → End(V ) be the projection. Then p 1 (S) = K s,d,n and p 1 : S → K s,d,n is a projective birational morphism. For s = d, K d,d,n is isomorphic to affine space and its defining ideal is generated by L ⊗ W ⊂ A 1 . For s < d, we can deduce from the map p 1 that the singular locus and non-normal locus of K s,d,n coincide and is K s+1,d,n , and that K s,d,n is an irreducible subvariety of codimension s(n − d) in End(V ) [OS,Theorem 4.4]. In particular, when n > d + 1, K s,d,n is not Cohen-Macaulay by Serre's criterion for normality. When s = 1 and n = d + 1, K 1,d,d+1 is a hypersurface and hence is Cohen-Macaulay, but we do not know what happens when s > 1 and n = d + 1 in general.

Characteristic-free multilinear algebra.
Given a partition λ = (λ 1 , . . . , λ n ), let ℓ(λ) be the largest i such that λ i = 0. If i λ i = n, we write λ ⊢ n and |λ| = n. The dual partition λ ′ is defined by λ ′ i = #{j | λ j ≥ i}. The notation a b means the sequence (a, . . . , a) (b times). Given a partition, we can think of it as a collection of boxes Let R be a commutative ring and let U be a free R-module of finite rank n. We define the determinant of U to be det U = n U . The Schur and Weyl functors are denoted L λ U and K λ U , respectively. See [Wey,Chapter 2] for their definition. However, we will change notation from [Wey, Chapter 2] so that we use L λ ′ U to mean L λ U . In particular, where S denotes symmetric powers and D denotes divided powers.
We recall the relevant properties that we need. First, both K λ U and L λ U are representations of GL(U ). Both K λ U and L λ U are free U -modules of the same rank, and this rank is given by [Sta,Corollary 7.21.4]. In particular, we have L λ (U ) = 0 if and only if ℓ(λ) > rank U and similarly with K λ (U ). Also, L λ 1 +1,...,λn+1 U = det U ⊗ L λ 1 ,...,λn U , and similarly for K, so we can use this to define L λ and K λ when λ is a weakly decreasing sequence of integers which are allowed to be negative. There is a canonical isomorphism L λ (U * ) = K λ (U ) * [Wey, Proposition 2.1.18]. Also there are isomorphisms L λ 1 ,...,λn (U * ) = L −λn,...,−λ 1 U and similarly for K [Wey,Exercise 2.18]. The functors L λ and K λ are compatible with base change. Hence it makes sense to construct L λ U and K λ U when U is a locally free sheaf on a scheme. When R is a Q-algebra (or Q-scheme), we have L λ U ∼ = K λ U . In this case, we will use the notation S λ U to make it clear that we are dealing with the characteristic 0 situation. In positive characteristic, they need not be isomorphic, and this is one of the reasons that some of our proofs will only be valid in characteristic 0.
Given two free modules U and U ′ , the symmetric powers S d (U ⊗ U ′ ) have a GL(U ) × GL(U ′ )equivariant filtration whose associated graded is Similarly, the exterior powers d (U ⊗ U ′ ) have a GL(U ) × GL(U ′ )-equivariant filtration whose associated graded is These are the Cauchy identities. Furthermore, given two partitions λ, µ, the tensor product L λ U ⊗ L µ U has a filtration whose associated graded is of the form [Bof,Theorem 3.7]. When R is a Q-algebra (or Q-scheme), the above filtrations become direct sum decompositions.
The c ν λ,µ are Littlewood-Richardson coefficients (see [Wey,Theorem 2.3.4]). We will only need to know these numbers when λ = (d) or λ = (1 d ), which we now explain. We say that ν is obtained from µ by adding a horizontal strip of length d if |ν| = |µ| + d and we have the inequalities ,µ is nonzero (and equal to 1) if and only if |ν| = |µ| + d and µ i ≤ ν i ≤ µ i + 1. These are the Pieri rules.

The geometric approach to syzygies.
Fix a field K. Let X be a projective variety and let U be a vector space, and denote the projections p 1 : U × X → U and p 2 : U × X → X. Let S ⊂ U × X be a subbundle with quotient bundle T and set Y = p 1 (S) ⊂ U . Also, set ξ = T * and let A = Sym(U * ) be the coordinate ring of U with grading given by deg U * = 1. The notation A(−i) denotes the ring A with a grading shift so that it is generated in degree i. For all i ∈ Z, define graded A-modules In particular, if the higher direct images of Sym(S * ) vanish and p 1 is birational, then F • is a minimal A-free resolution of the normalization of Y .
with the convention that negative exterior powers are 0. Now use the notation from Section 1.1. We consider the case of a Grassmannian X = Gr(s, L) whose points are the s-dimensional subspaces of L. The cotangent bundle of X is R ⊗ Q * .
Theorem 1.4 (Kempf vanishing). Let α, β be two partitions such that α d−s ≥ β 1 . Then Proof. For the first statement, see [BK,Theorem 3.1.1]. For the second statement, the sheaf Given a permutation w, we define the length of w to be ℓ Theorem 1.5 (Borel-Weil-Bott). Suppose that the characteristic of K is 0. Let α, β be two partitions and set ν = (α, β). Then exactly one of the following two situations occur.

There exists
and all other cohomology vanishes.

Normalizations of Kalman varieties.
Let O s,d,n denote the coordinate ring of K s,d,n and let O s,d,n denote the normalization of O s,d,n . In this section we prove some results on O s,d,n that will be used in the main results of this article (Theorem 3.3 and Theorem 3.6). Some additional results on the normalizations can be found in Proposition 3.8 and Proposition 3.10. Continue the notation of Section 1.1.
Proposition 2.1. Over a field of characteristic 0, the higher direct images of S vanish for all s, d, n. In particular, O s,d,n has rational singularities and hence is Cohen-Macaulay. The higher direct images also vanish in arbitrary characteristic in the case s = 1 and in the case s = 2, d = 3. In particular, O 1,d,n and O 2,3,n are flat over Z.
Combined with Theorem 3.2 we conclude that O 1,d,n and O 2,3,n are also flat over Z.
Proof. First suppose that characteristic is 0. By Theorem 1.3(a), it is enough to show that F i = 0 for i < 0. The summands of q ξ are of the form S λ R ⊗ S µ Q * ⊗ S ν W where |λ| = q and |µ| ≤ q. From the description of Borel-Weil-Bott (Theorem 1.5), it is clear that such a sheaf can only have cohomology in degree at most q, which proves the claim. Now suppose that the characteristic is arbitrary. For s = 1, the claim follows from Kempf vanishing (Theorem 1.4) since gr S = O + Hom(Q, V ) + Hom(W * , V ), so Sym(gr S * ) has no higher cohomology, and hence the same is true for Sym(S * ). The case of s = 2 and d = 3 will be shown in Proposition 2.5.
Remark 2.2. We expect that the higher direct images vanish for all s, d, n and in all characteristics, but we are unable to prove this. Proposition 2.3. O 1,d,n has (Castelnuovo-Mumford) regularity d − 1 and the terms of its minimal free resolution F • are Proof. Use the notation of Section 1.3. We have By Kempf vanishing (Theorem 1.4), the last term is 0 for j < d − 1. When j = d − 1, we get and this term contributes to F q−d+1 . The rest follows from Section 1.3.
Corollary 2.4. Let F • be the minimal free resolution of O 1,d,n . For i > 1, the only nonzero components in the differential F i → F i−1 are the maps with the convention that a term on the right is 0 if it does not appear in F i−1 .
Proof. Consider the Koszul complex of O S over the total space of End(V )×Gr(s, L). For simplicity, we work over Gr(s, L) by pushing forward along the projection (which is an equivalence since End(V ) × Gr(s, L) → Gr(s, L) is affine). The degree i + d − 1 component of the map above is obtained by applying H d−1 to the map of sheaves in this Koszul complex. The equations for O S are given by ξ = R ⊗ (Q * ⊕ W ) ⊂ End(V ). In particular, we can restrict our attention to the map .
Using Serre duality, this is the same as taking the dual map of applying H 0 to Since the differentials in the Koszul complex are obtained via comultiplication, both of these maps are given by exterior multiplication. Hence the map on sections is surjective, which implies that our desired maps are injective (and hence nonzero). That there are no other nonzero maps follows from Theorem 1.3(b).
Proposition 2.5. If the characteristic of K is 0, then the first few terms of the minimal free resolution F • of O 2,3,n are: The ranks of these F i are the same for any field. Furthermore, the regularity of O 2,3,n is 2.
Proof. Since dim Gr(2, 3) = 2, the regularity of O 2,3,n is at most 2 by Theorem 1.3(a). So the above reduces to calculating the cohomology of q ξ for 0 ≤ q ≤ 5, which we first do in characteristic 0. This is a straightforward, although tedious, application of the Cauchy identity, Pieri rule, and Borel-Weil-Bott theorem (all explained in Section 1.2), which we omit. Now assume that the field has characteristic p > 0. If p > 5, then we may still use Borel-Weil-Bott to calculate the cohomology of q ξ with q ≤ 5 (this reduces to the statement that the nth symmetric and divided power functors are naturally isomorphic when n! is invertible). In the remaining cases p ∈ {2, 3, 5}, the cohomology calculation can be reduced to a finite calculation with Macaulay 2 [GS], which we explain. First, we have ξ = R ⊗ (Q * ⊕ W ). Since we only go up to 5 ξ, we see that the terms which appear in the Cauchy filtration of i ξ are the same when dim W ≥ 5. So we only need to consider the case dim W = 5. Finally, we only need to calculate H 1 and H 2 since we know the Euler characteristic. For 5 ξ, we only care about H 2 . We use the following code: A=ZZ/2[z_0,z_1,z_2]; m=matrix{{z_0,z_1,z_2}}; R = sheaf((ker m) ** A^{1}); Q = sheaf(A^{1}); xi = (R ** dual(Q)) ++ (R ++ R ++ R ++ R ++ R); for i from 1 to 4 do ( E = exteriorPower(i,xi); print (rank HH^1(E), rank HH^2(E)); ) print rank HH^2(exteriorPower(5,xi));

Kalman varieties.
In this section we prove our main results, which include calculating the minimal free resolution of O 1,2,n and the equations of O 1,3,n . During the course of our work, we discovered the following conjecture. 2 ). There is a long exact sequence Furthermore, the ideal of O 1,d,n has minimal generators in degrees d, d+1, . . . , d(d+1)
The rest of the section will imply that this conjecture holds for d ≤ 3, so we record the result.
Theorem 3.2. Conjecture 3.1 holds when d ≤ 3. In particular, there are exact sequences For more precise statements about the number of equations, see Theorem 3.3 and Theorem 3.6. We expect that the methods used in these cases will extend to any given value of d, but we have been unable to properly organize the combinatorics in the case of general d. However, we are able to prove Conjecture 3.1 in the case n = d + 1 and char K = 0. We provide a brief sketch of this case in Section 3.4.
We point out that we were not able to check the conjecture computationally even for the first nontrivial case d = 4 and n = 6.

Syzygies for d = 2.
Theorem 3.3. The terms of the minimal free resolution F • of O 1,2,n are given by In particular, the projective dimension of O 1,2,n is 2n − 5 and it has regularity 2.
Proof. From Proposition 2.3, O 1,2,n has the following presentation: The map 2 L ⊗ 2 W ⊗ A(−2) → A(−1) is 0. We can either appeal to Theorem 1.3(b) or use that no such G-equivariant map exists. Hence the presentation for O 1,2,n /O 1,2,n must be L ⊗ W ⊗ A(−2) → A(−1), and we conclude that the quotient is O 2,2,n (−1). Let F • be the minimal free resolution of O 1,2,n from Proposition 2.3 and let G • be the Koszul complex on L ⊗ W resolving O 2,2,n (−1). We can lift the quotient map O 1,2,n → O 2,2,n (−1) to get a map of complexes F • → G • . The ith term of this map is We claim that the map from D i L ⊗ i W is an inclusion and the map from K i,1 L ⊗ i+1 W is 0. By minimality of F • , the map D i L ⊗ i W → F i−1 is injective, and by Corollary 2.4, the map is zero. By induction on i, we get the claim. Therefore we know exactly what the minimal cancellations in the comparison map F • → G • are, which gives the desired resolution F • via a mapping cone.
Remark 3.4. In the above proof, we know from general principles that the comparison maps F i → G i must be nonzero since both O 1,2,n and O 2,2,n are Cohen-Macaulay (see the proof of [BEKS,Proposition 2.3]). So one can deduce the required cancellations using just representation theory (at least in characteristic 0) without understanding the differentials.

Equations for s = 1 and d = 3.
In Proposition 2.5, we do not know how to write down the Z-forms for the representations of G involved, so we just switch to the notation (λ; µ) to mean some Z-form of the module S λ L ⊗ S µ W and we also write (−i) in place of ⊗A(−i).
Remark 3.7. Using (1.2), this proves [OS,Conjecture 3.6], which says that there are n−3 Proof. The proof is similar to that of Theorem 3.3. The presentation for O 1,3,n is is 0 since there are no nonzero such G-equivariant maps. Also, the maps from (1 2 ; 1 2 )(−3) and (1; 1)(−3) to A(−1) ⊕ A(−2) are nonzero. If not, then they give generators for the ideal of O 1,3,n . In particular, if we pick an ordered basis for V which first has a basis for L followed by a basis for W , then these equations correspond to the 2 × 2 minors and the 1 × 1 minors of the bottom-left block submatrix, respectively, and we can find matrices in K 1,3,n for which these equations do not vanish.
In this section, we assume that K has characteristic 0 and find the equations for O d−1,d,n . We can also do this in arbitrary characteristic when d = 3 since in this case, the next result is implied by Proposition 2.5.
Proof. Using arguments similar to before, the presentation for O d−1,d,n /O d−1,d,n is The maps in the above are of the form (1; 1)(−j −1) → A(−j) for j = 1, . . . , d−1. So in some choice of basis, the cokernel is d−1 j=1 O d,d,n (−j), which is resolved by a direct sum of Koszul complexes. The next term in the Koszul complex is d+1 j=3 [(1 2 ; 2)(−j)⊕(2; 1 2 )(−j)]. Let F • be the minimal free resolution of O d−1,d,n . Using arguments similar to before, all terms in F 2 of the form (1 2 ; 2)(−j) and (2; 1 2 )(−j) have a nonzero map to (1; 1)(−j + 1) in F 1 . Hence the maps from these terms to the corresponding terms of the Koszul complex of O d−1,d,n /O d−1,d,n are nonzero, and we finish the proof by a mapping cone construction.
In this section, we sketch a proof of Conjecture 3.1 in the case when char K = 0 and n = d + 1. Since the details are fairly involved and because this result is not very substantial, we will just mention the important points, and offer it as evidence for the validity of Conjecture 3.1.
Proposition 3.10. When char K = 0, the terms of the minimal free resolution F • of O s,d,d+1 are where λ ⊆ (s − i) × (d − s) means ℓ(λ) ≤ s − i and λ 1 ≤ d − s, and the empty partition is allowed.
Proof. We claim that the first s − 1 terms of the minimal free resolution F s • of C s are This hypothesis is just strong enough to allow one to prove the result and the claim by descending induction on s. The case s = d is clear since C d = O d,d,d+1 and is resolved by a Koszul complex.