On Rees algebras of 2-determinantal ideals

Let I be the ideal of minors of a 2 by n matrix of linear forms with the expected codimension. In this paper we prove that the Rees algebra of I and its special fiber ring are Cohen-Macaulay and Koszul; in particular, they are quadratic algebras. The main novelty in our approach is the analysis of a stratification of the Hilbert scheme of determinantal ideals. We study degenerations of Rees algebras along this stratification, and combine it with certain squarefree Groebner degenerations.


INTRODUCTION
Let M be a sufficiently general matrix of linear forms, and let I be its ideal of maximal minors.The study of blowup algebras associated to I, in particular the Rees algebra R(I) = ∞ k=0 I k , is a central subject in commutative algebra.For instance, it appears in the study of rational maps, special varieties, integral dependence, multiplicities, syzygies, and singularities of plane curves; see [P21] for an overview.In all these instances, one is naturally led to study the defining relations of R(I).In doing so, some of the most common problems involve computing the degrees of the defining relations, measuring the singularities of R(I), and investigating the Koszulness of R(I).These avenues of research are very active and we mention some of the more recent papers [ALL20, BC21, BCV15, CDFGLPS21, CP22, DMN21, L14, LS22a, LS22b, S20].
In this work, we provide a detailed picture for the case of matrices with two rows.Many interesting geometric objects are defined by minors of such matrices, for instance rational normal scrolls [EH87], 2-regular algebraic sets, small schemes [EGHP06], and eigenschemes of square matrices [AEKP16].Moreover, the rational maps associated with some of these objects exhibit rich geometry as seen in the work [RS01].For simplicity, we say that an ideal I is 2-determinantal if it is generated by the 2 × 2 minors of a 2 × (c + 1) matrix of linear forms, where c = codim(I).Our main result is the following: Theorem 1.1.Let I be a 2-determinantal ideal.The Rees algebra R(I) and the special fiber ring F(I) are Cohen-Macaulay Koszul algebras.In particular, they are defined by quadratic relations.
The theorem builds on and generalizes several previous results.When I defines a balanced rational normal scroll, it was proved in [CHV96] that R(I) and F(I) are Koszul and Cohen-Macaulay (in fact, that they have rational singularities).The results of [BCV15] show that the powers of any 2-determinantal ideal I have a linear free resolution, a property that is closely related to the Koszulness of the Rees algebra.Moreover, they show that I is an ideal of fiber type, i.e., the defining equations of R(I) consist of those of F(I) and the first syzgies of I.When I defines an arbitrary rational normal scroll, a part of the theorem was established in the works of [S20] and [LS22a].In [S20], the author proves the Koszulness of R(I) and F(I) by explicitly constructing Gröbner bases for their defining ideals.In [LS22a], the authors prove that F(I) is Cohen-Macaulay by showing that the simplicial complex determined by the Gröbner basis of [S20] is shellable.
The proof of Theorem 1.1 relies on a variety of techniques and, unlike the previously established results, has relatively little case analysis.The central idea is to seek convenient degenerations of Rees algebras of arbitrary 2-determinantal ideals that are algebraically simpler.This naturally leads one to examine stratifications of the Hilbert scheme of 2-determinantal ideals.The reason we believed that such an approach would be fruitful is due to a theorem of Harris [H81], which completely characterizes degenerations among scrolls by means of their integer partitions.Notably, Harris shows that, for each dimension and codimension, there exists a unique scroll, corresponding to the least balanced partition, such that every other scroll degenerates to it.Using the theory of Kronecker-Weierstrass normal forms, we investigate the analogous problem for the broader class of 2-determinantal ideals.For each dimension and codimension, we identify a distinguished 2-determinantal ideal L with the property that every 2determinantal (non-cone) ideal degenerates to it, see Theorem 4.3 for a more precise statement.The scheme defined by L is a union of two components, a smooth Segre variety and a non-reduced 2-regular scheme supported on a linear space, that are linearly joined along a codimension one linear space, in the sense of [EGHP06].
This fact, combined with results of [BCV15], reduces the proof of Theorem 1.1 to the study of the blowup algebras of the single ideal L, for each dimension and codimension.To conclude, we employ the combinatorial techniques developed in [S20] for the case of scrolls to prove that both F(L) and R(L) have a squarefree Gröbner basis of quadrics (Theorems 6.1 and 7.1).Although the proof outline is similar, the combinatorial analysis is substantially trimmer, chiefly because we only deal with one explicit ideal as opposed to all possible scroll partitions in [S20], and, thus, we obtain a shorter proof even for scrolls.
A more significant consequence of our analysis of degenerations concerns the Cohen-Macaulayness of the blowup algebras.This property was known to hold for both blowup algebras in the case of balanced scrolls [CHV96] via Sagbi bases, and for the special fiber ring of arbitrary scrolls [LS22a] via a detailed analysis of the initial complex.Combining our results with the theorem of Conca-Varbaro [CV20], we deduce effortlessly the Cohen-Macaulayness for both blowup algebras and for all 2-determinantal ideals.

PRELIMINARIES
Throughout this work, k denotes a field.In some sections, we assume that k is algebraically closed, but the main theorem will be proved for arbitrary fields.We use the symbol [•] d to denote a graded component of degree d.
2.1.Determinantal ideals.Let S = Sym(k n+1 ) be a polynomial ring and P n = Proj S its projective space.Given a matrix M, we denote by I t (M) the ideal generated by all the t × t minors of M. We fix a terminology for the ideals that are the main subject of this paper: Definition 2.1.An ideal I ⊆ S is called 2-determinantal ideal if there exists a 2 × (c + 1) matrix M of linear forms such that I = I 2 (M) and codim(I) = c.
Analogously, the closed subscheme V(I) ⊆ P n defined by a 2-determinantal ideal I ⊆ S is called a 2determinantal scheme.These objects enjoy many well-known properties.The condition that codim(I) = c means that I has the generic codimension with respect to the Eagon-Northcott bound, and this is equivalent to the fact that the Eagon-Northcott complex associated to M is a (minimal, linear) free resolution of S/I.It follows that S/I is a Cohen-Macaulay ring of (minimal) multiplicity c + 1, and, if I is prime and k = k, then V(I) is a variety of minimal degree.See [BV06] for details.Now, assume that k = k.For fixed integers c and n, all 2-determinantal schemes have numerically the same free resolution.Hence, they have the same Hilbert function and the same Hilbert polynomial p(ζ), and, as a consequence, they are parametrized by the same Hilbert scheme Hilb p(ζ) (P n ).We point out that there is a vast literature on Hilbert schemes of determinantal schemes, in a much more general setting, see e.g.[KMR20], but we will only be concerned with the 2-determinantal case.
2.2.Blowup algebras.Let I = I 2 (M) = (δ 1 , . . ., δ m ) ⊆ S be a 2-determinantal ideal, where the δ i are the 2 × 2 minors of M. The Rees algebra of I is the ring where τ is a variable.It is a standard bigraded ring by setting deg(x i ) = (1, 0), deg(τ) = (−2, 1), so that deg(δ i τ) = (0, 1).The special fiber ring of I is the subring F(I) ⊆ R(I) concentrated in bidegrees (0, * ), and it can also be identified with the subring k[δ 1 , . . ., δ m ] ⊆ S. The name "special fiber" comes from the fact that where ρ, ρ ′ are the algebra retractions defined by projecting onto bidegrees (0, * ), ι, ι ′ are the natural inclusions, and π R , π F are the (bi)graded surjective maps defined by π R (T i ) = δ i τ.The ideals ker(π R ) and ker(π F ) are called the defining ideals of R(I) and F(I), respectively.The ideal I is said to be of fiber type if ker(π R ) is generated in bidegrees ( * , 1) and (0, * ), equivalently, by the polynomials arising from the first syzygies of I and by ker(π F ).We refer to [P21] for an overview on defining ideals of blowup algebras.

Degenerations and singularities.
In this subsection, we assume k = k.We discuss the kind of flat families we will be primarily interested in.Let B be the localization k an open subset containing 0. We define a one-parameter family of graded algebras to be an N-graded B-algebra A such that [A] 0 = B and A is generated by [A] 1 .For α ∈ k, we denote by A α = A/(t − α) the corresponding member of the family.

Lemma 2.2. A one-parameter family A of graded algebras over B is flat if and only if all the members
A α have the same Hilbert function.
Proof.This can be found for example in [E95, Exercise 20.14].
Given standard graded k-algebras R, R ′ we say that R degenerates to R ′ if there exists a flat oneparameter family A such that A 0 ∼ = R ′ and A α ∼ = R for α = 0.By abuse of terminology, given homogeneous ideals I, I ′ ⊆ S, we say that I degenerates to I ′ if S/I degenerates to S/I ′ .A notable special case is the degeneration arising from a Gröbner basis [E95, Section 15.8].However, it is often fruitful to seek degenerations in the generality described above; in fact, this is one of the key aspects of this paper.
Several singularities of algebras behave well in flat families; we list the ones that are relevant to our main results.A standard graded k-algebra R is called Koszul if [Tor R i (k, k)] j = 0 whenever i = j, that is, if the field k has a linear free R-resolution.Koszul algebras are defined by quadrics, whereas algebras defined by Gröbner bases of quadrics are Koszul.See [CDR13] for more details on Koszul algebras.
A Noetherian local ring (R, m) is called cohomologically full if, for every local ring (T , n) such that char(R) = char(T ) and char(R/m) = char(T/n), and every surjection T → R such that the induced map A standard graded k-algebra R is called cohomologically full if its localization at the homogeneous maximal ideal is cohomologically full.See [DDM21] for details on cohomologically full rings.2) is well-known in the case of Gröbner degenerations, see e.g.[C09, Lemma 2.1], but the argument works for any one-parameter degeneration in the above sense.Similarly, item (3) is proved in [CV20, Proposition 3.3] in the case of square-free Gröbner degenerations, but the argument works for any one-parameter degeneration and any cohomologically full ring R ′ .
By [CV20], the converse of item (1) in Lemma 2.3 holds when R ′ is a Stanley-Reisner ring.More generally, we have: Proof.The statement follows from [CV20, Theorem 1.2].While the theorem is stated for Gröbner degenerations to a Stanley-Reisner ring R ′ , the argument works for any cohomologically full ring R ′ [CV20, Remark 2.6] and any one-parameter degeneration.

THE KRONECKER-WEIERSTRASS TYPE
The goal of this section is to study the structure of 2-determinantal ideals.In particular, we introduce a discrete invariant that we call the Kronecker-Weierstrass type, or simply KW type.It is based on the Kronecker-Weierstrass theory of matrices, which we review in this section.In order to define this invariant, we will describe the primary decomposition of 2-determinantal ideals.Throughout this section, we assume k = k.
Let S be a polynomial ring over k.A scroll block is a matrix of the form for some variables x 1 , . . ., x p+1 .A Jordan block is a matrix of the form where ε ∈ k is called the eigenvalue of the Jordan block.A nilpotent block is a matrix of the form where each M i is a scroll, Jordan, or nilpotent block and the sets of variables appearing in different blocks are disjoint.
Proposition 3.1.Let M be a 2 × (c + 1) matrix of linear forms in S. Using row and column operations and linear automorphisms of S, one can transform M into a matrix M ′ that is a concatenation of blocks.
Definition 3.2.Let I ⊆ S be a 2-determinantal ideal with codim(I) = c.A Kronecker-Weierstrass normal form of I is a 2 × (c + 1) matrix M that is a concatenation of blocks and satisfies ϕ(I) = I 2 (M) for some linear automorphism ϕ of S.
By Proposition 3.1, such a matrix M always exists.We emphasize that our definition includes the requirement that M has the least possible number of columns c + 1.For instance, the matrices are not Kronecker-Weierstrass normal forms, since their ideals of minors have codimension c = 2. Below, we classify all the Kronecker-Weierstrass normal forms of 2-determinantal ideals.
Proposition 3.3.Let M be a 2 × (c + 1) matrix that is a concatenation of blocks and let I = I 2 (M).Then, codim(I) = c if and only if one of the following two conditions holds: (1) M consists of scroll blocks and Jordan blocks with distinct eigenvalues.
(2) M consists of one nilpotent block.
Proof.This follows from [C90], where the Hilbert series is calculated for all concatenations of blocks.Specifically, the desired statement follows from [C90, Theorems 2.2.3 and 2.5.5].
Case (2) should be considered as an exceptional case in this classification.Ideals of this form are simply the squares of linear primes.Equivalently, they are the 2-determinantal ideals I 2 (M) such that codim I 2 (M) = codim I 1 (M).In this paper, we are concerned with the ideals of the form (1).
Kronecker-Weierstrass normal forms are not invariant under projective equivalence: the "continuous" data, i.e., the eigenvalues of the Jordan blocks, is not uniquely determined.A simple example is given by the concatenation of Jordan blocks whose ideal of minors is (x 1 x 2 , x 1 x 3 , x 1 x 4 , x 2 x 3 , x 2 x 4 , x 3 x 4 ) as long as the eigenvalues are all distinct.Another example is the following result, which will be useful later.
Lemma 3.4.Let M be a concatenation of scroll blocks and Jordan blocks.Given δ ∈ k, let M ′ denote the matrix obtained from M by adding δ to all the eigenvalues of the Jordan blocks.Then, I 2 (M) and I 2 (M ′ ) are the same ideal up to a change of coordinates involving only the variables of the scroll blocks.
On the other hand, we are going to see that the discrete data is invariant under projective equivalence.When M is a concatenation of scroll blocks, the ideal I = I 2 (M) is prime and V(I) is a variety of minimal degree, called a rational normal scroll.Lemma 3.5.Let M and M ′ be concatenations of scroll blocks of sizes Then, I 2 (M) and I 2 (M ′ ) are projectively equivalent if and only if d = d ′ and a i = a ′ i for every i.
Proof.This is well-known, see for instance [H81].
We now describe the primary decomposition of the ideals of the form (1) in Proposition 3.3.With a slight abuse of terminology, by multiplicity of an ideal I ⊆ S we mean the degree of the subscheme V(I), equivalently, the multiplicity of the ring S/I, and we denote this number by mult(I).
be a concatenation of scroll blocks S 1 , . . ., S d and Jordan blocks J 1 , . . ., J e with distinct eigenvalues.The ideal I = I 2 (M) has e + 1 primary components.One component is the prime ideal , which defines a rational normal scroll.For each 1 i e, there is a component q i primary to a linear prime and with multiplicity equal to the number of columns of J i .
Proof.Assume M has c + 1 columns.Since codim(I) = c by Proposition 3.3, the ring S/I is Cohen-Macaulay of multiplicity c + 1.It follows that I is unmixed, i.e. all associated primes are minimal and have codimension c, and the multiplicities of the primary components add up to c + 1.
We fix a notation for the variables in the blocks of M by setting In particular, we have c + 1 = d i=1 a i + e i=1 b i .Up to adjoining variables, we may assume that the ambient polynomial ring is S = k[x 1,1 , . . ., x d,a d +1 , y 1,1 , . . ., y e,b e ].
Consider the ideals They are the defining ideals of a rational normal scroll and a linear space, respectively.It follows that they are both prime and that codim( b i , and mult(J ′′ ) = 1.
We claim that, for every Jordan block J i , there is a linear minimal prime p i of I such that I 1 (J i ) ⊆ p i and I 1 (J h ) ⊆ p i for all h = i, and such that the p i -primary component q i has mult(q i ) = b i .This claim implies the conclusion of the theorem.In fact, it follows from the claim that the minimal primes p 0 , . . ., p e are all distinct.Moreover, since the sum of the multiplicities of all the primary components of I must equal mult(I) = d i=1 a i + e i=1 b i , we can conclude that mult(q 0 ) = d i=1 a i , and hence q 0 = p 0 .This shows that q 0 , q 1 , . . ., q e are the only primary components.
It suffices to prove the claim for one Jordan block, say J 1 .Applying Lemma 3.4, we may assume that its eigenvalue is ε 1 = 0. Let p 1 be the linear prime generated by the second row of M, equivalently, x i,2 , . . ., x i,a i +1 + y 1,2 , . . ., y 1,b 1 + e i=2 y i,1 , . . ., y i,b i .
Then codim(p 1 ) = d i=1 a i + (b 1 − 1)+ e i=1 b i = c = codim(I) and I = I 2 (M) ⊆ p 1 , so p 1 is a minimal prime.It remains to show that mult(q 1 ) = b 1 , equivalently, that the Artinian ring (S/I) p 1 has length b 1 .Observe that K = k(x 1,1 , x 2,1 . . ., x d,1 , y 1,1 ) ⊆ (S/I) p 1 .We are going to show that thereby concluding the proof.Let a = d i=1 a i be the number of scroll columns.The variable y 1,1 , in position (1, a + 1) in M, is invertible in S p 1 .Using row and column operations we reduce M to 0 where ∆ i is the 2 × 2 minor of M involving columns i and a + 1.Thus, we have It is convenient to relabel the minors with respect to variables and blocks they appear in.Let We have For each scroll block S i we may rewrite For the Jordan block J 1 we have For the Jordan blocks J i with i > 1 we consider the unit γ i = (z − ε i ) ∈ S p 1 and we have Thus, we may apply a local change of coordinates in the ring S p 1 carrying The desired conclusion (3.1) follows from (3.2).
It follows from Proposition 3.3, Lemma 3.5, and Theorem 3.6 that the Kronecker-Weierstrass normal form of a 2-determinantal ideal I is uniquely determined, up to the eigenvalues and rearrangement of the blocks.Thus, we may give the following definition of Kronecker-Weierstrass type.In this paper, an integer partition is a sequence of integers λ = (λ 1 , . . ., We use the exponential notation to denote repetitions, for example, (3 2 , 2, 1 4 ) = (3, 3, 2, 1, 1, 1, 1).Definition 3.7.Let λ = (λ 1 , . . ., λ d ) and µ = (µ 1 , . . ., µ e ) be integer partitions.An ideal I ⊆ S has Kronecker-Weierstrass type (λ; µ) if there is a linear automorphism ϕ of S such that ϕ(I) scroll blocks S i of size λ i and Jordan blocks J i of size µ i with distinct eigenvalues.An ideal I ⊆ S is of nilpotent Kronecker-Weierstrass type if it is the square of a linear prime.
We will often abbreviate the phrase "Kronecker-Weierstrass type" to "KW type".We point out that the Kronecker-Weierstrass type is not a complete invariant of projective equivalence.In other words, unlike the case of rational normal scrolls, there exist 2-determinantal ideals with the same KW type that are not projectively equivalent.
Example 3.8.We consider ideals of KW type (1, 1; 1 n ).For each set e = {ε 1 , . . ., ε n } of distinct eigenvalues, consider the Kronecker-Weierstrass normal form The ideal I(e) = I 2 (M(e)) has codimension n + 1 and multiplicity n + 2. By Theorem 3.6, it has n + 1 primary components, and they are all prime.In fact, from the proof we see that they are The linear span of the quadric surface V(p 0 ) is P 3 = V(y 1 , . . ., y n ).Consider the lines and their collection L(e) = ℓ 1 (e), . . ., ℓ n (e) .As e varies, L(e) describes a locus L ⊆ Sym n Gr(1, P 3 ) of dimension n.Now, suppose there exists a linear automorphism ϕ of S such that ϕ(I(e)) = I(e ′ ) for two tuples e, e ′ .Then ϕ must fix the scroll component V(p 0 ) and, hence, its linear span, so it restricts to a linear automorphism ϕ of P 3 .Moreover, ϕ must carry the components {p 1 (e), . . ., p n (e)} to {p 1 (e ′ ), . . ., p n (e ′ )}, therefore, ϕ must carry L(e) to L(e ′ ).If n ≫ 0, it is clear, by dimension considerations, that L cannot be a single GL 4 (k)-orbit.Thus, for general e, e ′ , there will be no such linear automorphism ϕ.
Remark 3.9.The work [AEKP16] contains results related to the content of this section.Roughly speaking, the authors describe the primary decomposition of the determinantal ideals of a matrix consisting only of Jordan blocks, but with possibly repeated eigenvalues.

DEGENERATIONS OF 2-DETERMINANTAL IDEALS
Let S = Sym(k n+1 ) be a polynomial ring over k = k.All 2-determinantal ideals of S of a fixed codimension c are parametrized by points of the same Hilbert scheme Hilb p(ζ) (P n ), see Section 2. It is therefore natural to investigate degenerations among 2-determinantal ideals: when does a given 2-determinantal ideal degenerate to another one?In particular, for applications to blowup algebras, singularities, defining equations, etc., it is desirable to identify a set of "most degenerate" 2-determinantal ideals in the Hilbert scheme.There is a straightforward answer to the latter problem: the revlex generic initial ideal of any 2-determinantal ideal is of the form (x 1 , . . ., x c ) 2 , which is itself 2-determinantal, of nilpotent KW type.However, this answer is not satisfactory, because any such degeneration will usually fail to induce a degeneration of the blowup algebras.
We are interested in degenerations of 2-determinantal ideals I = I 2 (M), not of nilpotent KW-type, with the additional requirement that codim I 1 (M) is constant throughout the degeneration.Up to adjoining variables, the problem reduces to the case where codim that is, where the entries of M generate the maximal ideal of S. In terms of KW type, this amounts to the fact that the number of scroll blocks is equal to It follows that each L (λ;µ) is irreducible, and that it is a constructible subset of Hilb p(ζ) (P n ), i.e., it is a disjoint union of locally closed subsets.Moreover, by Theorem 3.6 and Definition 3.7, the space H c,d is equal to the disjoint union of all the L (λ;µ) , in other words, the loci L (λ;µ) form a stratification of H c,d .In order to answer the questions stated at the beginning of this section, we will study the poset defined by the relation L (λ;µ) ⊆ L (τ;ρ) .
A classical theorem of Harris gives a complete solution for the case of scrolls.It follows that, if we restrict to scrolls, there is a unique most degenerate rational normal scroll in each H c,d , namely, the one with the least balanced partition (c − d + 2, 1 d−1 ).
Our main result in this section is the existence of a unique most degenerate 2-determinantal ideal in each H c,d , up to linear changes of coordinates.In order to prove Theorem 4.3, we construct two basics degenerations.
Lemma 4.4.Let λ be a partition, and let λ ′ be a partition obtained from λ by replacing one part λ j > 1 with λ j − 1.An ideal with KW type (λ; µ) degenerates to an ideal with KW type (λ ′ ; µ, 1).Moreover, we have be a concatenation of scroll blocks S i of size λ i and Jordan blocks J i of size µ i with distinct eigenvalues ε i .For simplicity, we rename p = λ j .Consider the j-th scroll block and We claim that the family of ideals I 2 (M(t)) ⊆ S[t] is flat, and that the KW type of I 2 (M(t)) is (λ; µ) for t = 0 and (λ ′ ; µ, 1) for t = 0.For t = 0, the change of coordinates x p+1 → ε −1 x p+1 transforms S j (0) into a concatenation of a scroll block of size p − 1 and a Jordan block of size 1 with eigenvalue ε −1 .Thus, I 2 (M(0)) has the desired KW type (λ ′ ; µ, 1).
For t = 0, the change of coordinates .
Denoting by C i the i-th column of (4.1), we perform the column operations C i → tC i + εC i+1 for i = p − 1, p − 2, . .., 1, transforming the block (4.1) into the original scroll block S j .We conclude that I 2 (M(t)) has KW type (λ; µ) for all t = 0.The flatness of the family I 2 (M(t)) follows from Lemma 2.2: all ideals in the family have the same Hilbert function since they are 2-determinantal ideals with the same codimension (Proposition 3.3).
The last statement of the lemma follows by the same argument.Indeed, given any 2-determinantal ideal I ∈ L (λ ′ ;µ,1) , we may assume, by Lemma 3.4, that all the eigenvalues in a Kronecker-Weirstrass normal form are non-zero.Now, the argument above shows that I is a flat limit of ideals in L (λ;µ) .Lemma 4.5.Let µ be a partition, and let µ ′ be a partition obtained from µ by replacing two parts µ h , µ k with one part µ h + µ k .An ideal with KW type (λ; µ) degenerates to an ideal with KW type (λ; µ ′ ).Moreover, we have be a concatenation of scroll blocks S i of size λ i and Jordan blocks J i of size µ i with distinct eigenvalues ε i .By Lemma 3.4, we may assume ε h = 0, and for simplicity we rename ε = ε k , p = µ h , and q = µ k .Consider the two Jordan blocks We modify them setting be the corresponding concatenation.As in Lemma 4.4, we claim that I 2 (M(t)) ⊆ S[t] is a flat family yielding the conclusion of the lemma.For t = 0, the ideal I 2 (M(0)) is clearly of KW type (λ; µ ′ ), since the matrix ) is a single Jordan block with eigenvalue 0 and size p + q.Now let t = 0.By repeatedly subtracting multiples of the columns of J k (t) from the last column of J h (t), we transform the matrix J h (t) into Again, subtracting multiples of the columns of J k (t) from the second last column of this matrix we get Repeating this process of changing coordinates and column operations, we see that we can transform the matrix J h (t) into A final change of coordinates x 1 → x 1 − q i=1 α ′′ i y i restores the original block J h .We conclude that I 2 (M(t)) is of KW type (λ; µ) for general t ∈ k, that is, for t ∈ k such that t = 0 and tε = ε i for all i = h, k.Flatness follows as in Lemma 4.4 by Lemma 2.2 and Proposition 3.3.
The last statement of the lemma also follows as in Lemma 4.4: for any I ∈ L (λ;µ ′ ) , we may assume, by Lemma 3.4, that the eigenvalue corresponding to the Jordan block of size µ h + µ k is 0, and the argument above shows that I is a flat limit of ideals in L (λ;µ) .We now discuss the main application of the results of this section to the study of blowup algebras.In general, given a flat family of ideals, their Rees algebras or the special fiber rings need not form flat families.However, this is turns out to be true for 2-determinantal ideals I 2 (M), as long as the number codim I 1 (M) is constant along the family.Corollary 4.7.Let I, J ∈ H c,d be ideals such that I degenerates to J.Then, R(I) degenerates to R(J) and F(I) degenerates to F(J).

Proof of
Proof.A one-parameter family of ideals, as defined in Section 2, determines a one-parameter family of their Rees algebras, likewise for the special fiber rings.The claim follows from Lemmas 4.6 and 2.2.While Theorem 4.3 and Corollary 4.7 are sufficient for our investigation of the blowup algebras, it is natural to ask for the full poset of degenerations among KW types in H c,d , that is, the complete extension of Theorem 4.2 to all 2-determinantal ideals.It is possible to show that this poset is precisely the one generated by the degenerations of Theorem 4.2, Lemma 4.4, and Lemma 4.5.In other words, an ideal of KW type (λ 1 , . . ., λ d ; µ 1 , . . ., µ p ) degenerates to one of KW type (λ ′ 1 , . . ., λ ′ d ; µ ′ 1 , . . ., µ ′ q ) if and only if there exist a 1 , . . ., a d ∈ N such that • the partition µ ′′ = (µ 1 , . . ., µ p , 1 a ) is a refinement of µ ′ , in the sense that there exists a set partition P 1 , . . ., P q of {1, . . ., p + a} such that µ ′ j = i∈P j µ ′′ i for all j = 1, . . ., q.
Again, the condition is sufficient by Theorem 4.2 and Lemmas 4.4, 4.5, whereas a formal proof of the necessity would take us too far away from the scope of this paper and therefore we omit it.We end the section by illustrating the case of H 6,3 , i.e., the poset of degenerations of a 3-dimensional balanced rational normal scroll of degree 7.

BLOWUP ALGEBRAS OF THE MOST SPECIAL DETERMINANTAL IDEALS
The results of Section 4 reduce the proof of Theorem 1.1 from arbitrary 2-determinantal ideals to a single ideal in each H c,d , of KW type (1 d ; c − d + 1).Our next goal is completing this task.In this section, we set up the notation for the blowup algebras, and introduce four classes of polynomial relations and a monomial order for them.
Since we are only concerned with one KW type from now on, we fix a more convenient notation.
Thus, L is a 2 × (c + 1) matrix in Kronecker-Weierstrass normal form and L is a 2-determinantal ideal with KW type (1 d ; e) and codimension c.
Remark 5.2.The first three classes of relations are well-known, and they exist in general for all ideals of maximal minors.The Eagon-Northcott relations are the first syzygies in the Eagon-Northcott complex.The Plucker relations are the defining relations of the Grassmannian variety.The relations of the fourth class are less well-known, as they arise from the special structure of the matrix L; however, similar relations have appeared in [CHV96] and [S20] in the case of rational normal scrolls.
5.2.The monomial order.We now introduce a monomial order on the polynomial rings P R = k[y i,j , x h , T α,β ] and, by restriction, P F = k[T α,β ].This order, while apparently convoluted, has the benefit of determining two well-behaved initial complexes for R(L) and F(L), specifically, two flag complexes with good homological and combinatorial properties, as we will see in the next sections.The definition of is not particularly important: what matters are the initial terms it picks from the polynomial relations described above.
The order is constructed by defining a hierarchy of rules for breaking ties among monomials.
In other words, mdeg 1 (•) keeps track of columns in the matrix L; for the variable x α , it selects the column on which it appears in the first row.In the second step, we compare monomials according to their multidegree in N c+1 .
(5.10) Proof.In order to verify the statements, we need to compare the terms in the support of each polynomial following the four steps in the construction of until we obtain a unique highest monomial.
Step (1) is only relevant for polynomials containing some y 2,i , that is, for LEN α,β,γ with α d, and for them the unique highest term is the desired y 2,α T β,γ .
In step (2), we note that Plucker and upper Eagon-Northcott relations are homogeneous with respect to mdeg 1 (•), so this step has no effect on them, while the remaining relations are not homogeneous.In both cases, δ c and δ = c + 1, the unique highest monomial in LAP α,β,γ,δ is T α+1,β+1 T γ,δ as desired.Likewise, for each LEN α,β,γ with α d + 1 we see that x α+1−d T β,γ is the unique highest monomial.

EQUATIONS OF THE SPECIAL FIBER RING
In this section, we prove the following theorem: Theorem 6.1.The Plucker and Laplace relations (5.4), (5.5), (5.6) form a Gröbner basis of the defining ideal ker(π F ) of the special fiber ring F(L) with respect to .
The proof strategy is very similar to that of [S20, Section 3], based on the enumerative analysis of the initial simplicial complex of F(L).The analysis is substantially trimmer here, since we only consider the KW type (1 d ; e), whereas in [S20] a different complex is needed for every scroll partition.Rather than giving a completely self-contained treatment, we present the full analysis but refer to [S20] for the combinatorial calculations of Proposition 6.4.The (new) case d = 0 requires the additional computation of the multiplicity of F(L), which we perform in Appendix A.
We refer to [HH11] for background on monomial ideals, simplicial complexes, and the Stanley-Reisner correspondence.
Let ∆ F be the flag simplicial complex defined by the squarefree quadratic monomials (5.9) and (5.10), that is, the complex whose Stanley-Reisner ideal is The relation of inclusion between intervals makes the vertex set into a poset (V, ⊆).If a subset F ⊆ V satisfies ( †), then the Hasse diagram of the poset (F, ⊆) is a graph without cycles, hence a union of (ordered) trees.We use tree-related terminology such as root, children, leaves, and internal nodes.
When Laplace relations are absent, i.e. when e 3, ∆ F is the non-crossing complex of the Plucker algebra [PPS10], and the numerical data of its facets is easy to determine:

Every facet contains 2c − 1 intervals, and the number of facets is
Proof.Note that ( ‡) is vacuous when e 3. The conditions follow quickly from the fact that a facet is a subset F ⊆ V that satisfies ( †) and is maximal with respect to this property.Thus, facets correspond to full binary trees on c leaves.It is well-known that such tree have 2c − 1 nodes, and that the number of such trees is the Catalan number When Laplace relations are present, and F ⊆ V is a subset satisfying ( †), then ( ‡) holds if and only if it holds for the leaves of F. This yields an analogous description of the facets of ∆ F .

and only if the Hasse diagram of the poset (F, ⊆) satisfies the following conditions:
• F is a binary tree with root (1, c + 1); • if (α, β) ∈ F has two children, then they are of the form (α, γ), (γ, β) for some α < γ < β; Theorem 6.1 is now a consequence of the following criterion.It is a well-known application of the associativity formula for multiplicities, but we include a proof since we could not find one in the literature.A simplicial complex is called pure if all its facets have the same cardinality.Lemma 6.5.Let R be a polynomial ring, a monomial order, and K ⊆ R a homogeneous ideal.Let ∆ be a pure simplicial complex whose Stanley-Reisner ideal I(∆) ⊆ R satisfies I(∆) ⊆ in (K).Suppose that the rings R/K and R/I(∆) have the same Krull dimension and multiplicity.Then, I(∆) = in (K).
Proof.Denote J = in (K).Then, R/J and R/I(∆) also have the same Krull dimension and multiplicity.Since I(∆) ⊆ J, the equality I(∆) = J can be checked locally at the associated primes of the smaller ideal I(∆).Since all facets of ∆ have the same size, all associated primes of I(∆) have the same height.Since J and I(∆) have equal height, the associated primes of J of minimal height are also associated primes of I(∆).Finally, since R/J and R/I(∆) have the same multiplicity, the associativity formula for multiplicities [E95, Exercise 12.11.e]implies that I(∆) P = J P for all P ∈ Ass(I(∆)).
Proof of Theorem 6.1.The special fiber ring F(L) and the Stanley-Reisner ring P F /I(∆ F ) have the same dimension and multiplicity.For P F /I(∆ F ), these invariants are found in Propositions 6.2, 6.3, 6.4,where it is also shown that ∆ F is pure.For F(L), they are found in Propositions A.1 and A.2.Moreover, I(∆ F ) ⊆ in (ker(π F )) by Proposition 5.3.Now, apply Lemma 6.5 to conclude.
The strategy we adopt here is different from the one of Section 6, for two reasons: the initial complex of R(L) is considerably more convoluted than that of F(L), and a formula for the multiplicity of R(L) is not explicitly known.First, we will use the criterion of Lemma 6.5 to verify the special case d = 1, where the simplicial complex is relatively simple and the multiplicity of R(L) can be derived from results on rational normal curves [C00].This is enough to deduce the general case, thanks to the fiber type property [BCV15] and the observation the all S-pairs reductions follow from the special cases d = 1, c 5, or Theorem 6.1 (with one exception).
7.1.The case d = 1.In this subsection, we use Notation 5.1 where additionally we fix d = 1, so c = e.For simplicity, we rename y 1,1 = y 1 , y 2,1 = y 2 , so L is the ideal of minors of Let ∆ R be the flag simplicial complex defined by all the squarefree quadratic monomials determined in Proposition 5.3, that is, the complex whose Stanley-Reisner ideal is Proposition 7.2.If e 4 then ∆ R has 2 e+2 − (e + 1) 2 − 3 facets, each containing e + 3 elements.
Proof.Let W = {y 1 , y 2 , x 1 , . . ., x e , T 1,2 , T 1,3 , . . ., T e,e+1 } be the vertex set of ∆ R .Since y 1 and T 1,2 do not appear in any generator of I(∆ R ), they belong to every facet.If a face F contains x h for some h > 1, then F does not contain any T α,β with 1 α < β h and (α, We will now count the facets according to 5 different cases.Let F be a facet.Case 1: y 2 ∈ F.Then, T β,γ / ∈ F for all 2 β < γ e + 1, so any T -vertex appearing in F must be We conclude that the facets are of the form {y 1 , y 2 , T 1,2 , x 2 , . . ., x β−1 , T 1,β , . . ., T 1,e+1 } for 3 β e + 1 and {y 1 , y 2 , T 1,2 , x 1 , . . ., x e }.This gives a total of e facets, all of size e + 3. For the remaining cases we assume y 2 / ∈ F. Case 3: x 1 ∈ F and x h / ∈ F for every h > 1.Then, T 1,γ / ∈ F for all 3 γ c.For any T α,β ∈ F with (α, β) = (1, 2) we have (α, β) ⊆ (2, e + 1); it follows that F = {x 1 , y 1 , T 1,2 } ∪ F ′ where F ′ corresponds to a facet of ∆ F for the KW type (∅; e).Again, by Proposition 6.3 all such facets F have size e + 3, and by Proposition 6.4 their number is e−2 h=1 e−1 h − (e − 2) = 2 e−1 − 2 − (e − 2) = 2 e−1 − e. Case 4: x 1 / ∈ F and x h ∈ F for some h > 1.As noted in the beginning, the x-vertices contained in F are x a+1 , . . ., x a+b , for some 1 b e − 1, 1 a e − b.Then, for each T α,β ∈ F \ {T 1,2 } we have α a + 1 and β > a + b, and by condition ( †) in Section 6 the intervals (α, β) form a (saturated) chain with respect to inclusion.Counting the number of such chains is simple: we start at T 1,e+1 , then we choose either T 2,e+1 or T 1,e , and so on, until we choose T α,β such that β − α = b.For each b we make e − b choices, and we exclude the set of choices Case 5: x 1 ∈ F and x h ∈ F for some h > 1.The x-vertices in F are now x 1 , x a+1 , x a+2 , . . ., x a+b for some 1 b e − 1, 1 a e − b.We argue as in Case 4 to count these facets, the main difference is that T 1,e+1 / ∈ F and instead we start from T 2,e+1 .This time we have there are 2 e−1−b sets of choices for each 1 b e − 1, and all of them are valid, for a total of e−1 b=1 2 e−1−b = 2 e−1 − 1 facets.Trading T 1,e+1 for x 1 , we see again that all facets have e + 3 elements.
Proof of Theorem 7.1 for d = 1, e 4. It follows from Propositions 7.2, A.3, and Lemma 6.5.7.2.The general case.In this subsection, we complete the proof of Theorem 7.1.
Proof.It follows from Theorem 6.1, the fiber type property [BCV15, Theorem 3.7], and the well-known fact that the Eagon-Northcott relations generate all the first syzygies of a 2-determinantal ideal.Thus, by Buchberger's criterion, it suffices to show that all S-pairs reduce to 0. It is known that an S-pair S(F, G) reduces to 0 if gcd LM(F), LM(G) = 1.Since ker(π R ) is generated by quadrics, we then only need to analyze linear S-pairs, i.e.S-pairs of the form S(F, G) = aF − bG where a, b are terms of degree 1.We are going to show that various subsets of the generators of ker(π R ) are themselves Gröbner bases; we have already seen one instance in Theorem 6.1.
With the help of [M2], we verify the statement when c = 5, that is, for the matrices of KW types (1 6 ; ∅), (1 5 ; 1), (1 4 ; 2), (1 3 ; 3), (1 2 ; 4), (1; 5), (∅; 6).We perform the computation over the base ring k = Z, which implies it for any field of any characteristic.But these 7 KW types imply the arbitrary case, since they contain all possible column configurations for linear S-pairs.More specifically, consider the arbitrary case, we show that any S-pair in the set G reduces to 0 modulo G.By Theorem 6.1, we do not need to consider S-pairs among Plucker relations.If we have a linear S-pair S(PLU α,β,γ,δ , UEN ε,φ,ψ ), then the T -variable in LM(UEN ε,φ,ψ ) also appears in LM(PLU α,β,γ,δ ), so the number p of distinct column indices among α, β, γ, δ, ε, φ, ψ is either p = 4 or p = 5.Since the variable x 1 behaves differently from the others in the monomial order , we need to distinguish whether or not the column d + 1 appears among the p indices.In any case, this configuration of columns of L already occurs for one of the 7 matrices L ′ with 6 columns.Since the claim holds for c = 5, there exists an equation of reduction of the S-pair to 0 for L ′ , and this equation can be chosen to be homogeneous with respect to mdeg(•), in other words, containing only relations involving the p column indices.The same equation holds for the original p columns of L, so S(PLU α,β,γ,δ , UEN ε,φ,ψ ) reduces to 0 modulo G.The case of the linear S-pair S(UEN α,β,γ , UEN ε,φ,ψ ) is completely analogous.
Note that the reason we included the third column in L ′ is to guarantee that the monomial order for L ′ is a restriction of the one for L 3,7 , since x 1 behaves differently from x 2 , . . ., x e , cf.Subsection 5.2.
Proof.The proof proceeds exactly as the one of Lemma 7.4, the only modifications being the column multigrading, where we set mdeg(x α ) = e α+d−1 instead of mdeg(x α ) = e α+d , and the fact that the distinguished column is not d + 1 but c + 1, since its entry on the second row is 0.
The simple argument of Lemmas 7.4, 7.6 cannot be applied to S-pairs involving Laplace relations, or to S-pairs involving both upper and lower Eagon-Northcott relations, due to the absence of a suitable column multigrading.In fact, the reduction of these S-pairs will typically extend beyond the columns originally involved in the S-pair.However, we can reduce all but one of these to the case d = 1.
Proof.There are three kinds of S-pairs of bidegree (1, 2): those of the form S(LAP α,β,γ,δ , UEN ε,φ,ψ ) and S(LAP α,β,γ,δ , LEN ε,φ,ψ ), where the two leading monomials share a T -variable, and the case of S(LEN α,β,γ , UEN ε,φ,ψ ) where gcd LM(LEN α,β,γ ), LM(UEN ε,φ,ψ ) = x h for some h.Inspecting Proposition 5.3, we see that, in each case, at most one scroll column is involved in the S-pair, while all the other columns belong to the Jordan block.It follows that this configuration of columns of L already occurs in a matrix L ′ with KW type (1; 1 e ) for some e ≫ 0. By Subsection 7.1, there exists an equation of reduction of the S-pair to 0 for L ′ , and the same equation holds for the original S-pair in L.
To summarize, the highest monomial in either side of (7.1) occurs exactly once, respectively in LEN ε,φ,ψ and in UEN α,β,γ .The second highest monomial also occurs once; in the right-hand side it occurs in UEN α,β,γ , while in the left-hand side it does not occur in LEN ε,φ,ψ .Rearranging the addends of (7.1) we obtain an equation of reduction for S(UEN α,β,γ , LEN ε,φ,ψ ): the highest monomial cancels in the S-pair, while the second highest monomial does not cancel in the S-pair and becomes the leading monomial, dominating all the remaining terms in the equation.
Proof of Theorem 7.1.By Theorem 6.1 and Lemmas 7.7, 7.8, we see that all S-pairs reduce to 0.

COHEN-MACAULAYNESS AND PROOF OF THE MAIN THEOREM
The degenerations of blowup algebras established in Section 4 allow one to deduce their Cohen-Macaulayness essentially for free.In fact, combining with the results of Sections 6, and 7, we deduce that the blowup algebras of any I ∈ H c,d admit a sequence of degenerations to Stanley-Reisner rings.Since Stanley-Reisner rings are cohomologically full, it follows from Lemma 2.4 that if some algebra in this sequence is Cohen-Macaulay, then every algebra is Cohen-Macaulay.For d > 0, we may use the results on balanced rational normal scrolls from [CHV96], while the case d = 0 follows easily from d = 1, thanks to the following combinatorial observation.
Denote by ∆ R (d, e), resp.∆ F (d, e), the initial complex of the Rees algebra R(L), resp.special fiber ring F(L), associated to the KW type (1 d ; e).Given a face F of a simplicial complex ∆, the link of Lemma 8.1.The complex ∆ R (0, e) is isomorphic to a link of ∆ R (1, e), and ∆ F (0, e) is isomorphic to a link of ∆ F (1, e).
First, we observe that x 1 and T 2,e+1 appear in every facet of ∆ R (0, e), since they do not appear in the generators of the Stanley-Reisner ideal I ∆ R (0, e) by Theorem 7.1 and Proposition 5.3.Now, consider the face F = {x 1 , T 2,e+1 } ∈ ∆ R (1, e).Inspecting the generators of I ∆ R (1, e) , or the description of the facets in Proposition 7.2, we see that if ∈ F ′ for γ 3, and y 1 ∈ F ′ but y 2 / ∈ F ′ .In other words, F ′ ⊆ {y 1 , T 1,2 } ∪ G for some G ∈ ∆ R (0, e).
Conversely, for every G ∈ ∆ R (0, e) we have G ∪ {y 1 , T 1,2 } ∈ ∆ R (1, e), since y 1 , T 1,2 belong to every facet of ∆ R (1, e).We conclude that link F by exchanging y 1 , T 1,2 with x 1 , T 2,e+1 .Restricting to only the T -vertices we obtain the statement for ∆ F (0, e) and ∆ F (1, e).Now, we are ready to combine all the results of the paper and prove our main theorem.Then, there is a balanced rational normal scroll J ∈ H c,d .Applying Lemmas 4.4 and 4.5, we find a sequence of ideals {I (i) } q i=1 in H c,d such that I (1) = J, I (p) = I, I (q) = L for some 1 p q, and I (i) degenerates to I (i+1) for all i.By Corollary 4.7, R(I (i) ) degenerates to R(I (i+1) ) for all i.By Theorem 7.1, there is a further degeneration of R(I (q) ) to the quadratic Stanley-Reisner ring S of the simplicial complex ∆ R (d, e).Since S is Koszul by [CDR13, Remark 6], we can apply Lemma 2.3 (2) repeatedly and obtain that R(I (i) ) is Koszul for every i.Since S is cohomologically full by [DDM21, Remark 2.5], we can apply Lemma 2.3 (3) repeatedly and obtain that R(I (i) ) is cohomologically full for every i.Since R(I (1) ) is Cohen-Macaulay by [CHV96, Theorem 3.8], we can apply 2.4 repeatedly and obtain that all R(I (i) ) and S are Cohen-Macaulay.The same argument works for the special fiber rings.
Finally, assume that d = 0, that is, c = n.In this case, there are no rational normal scrolls in H c,0 .By the previous paragraph, the complexes ∆ R (1, e) and ∆ F (1, e) are Cohen-Macaulay.By Lemma 8.1 and [HH11, Corollary 8.1.8],it follows that ∆ R (0, e) and ∆ F (0, e) are also Cohen-Macaulay.Now, applying Lemma 2.3 as in the previous paragraph, it follows that R(I) and F(I) are Koszul, Cohen-Macaulay, and cohomologically full for every I ∈ H c,d .
We record some of the facts we proved on the way to proving Theorem 1.1.As another byproduct, Theorem 1.1 also implies that the formulas for the reduction number of I found in [CHV96, Corollary 4.8], in the case of balanced scrolls, are valid for any 2-determinantal ideal.In fact, if F(I) is Cohen-Macaulay, then the reduction number of I coincides with the degree of the numerator of the Hilbert series of F(I).The reduction number plays an important role in the theory of residual intersections, see e.g.[EHU20,EU22] for recent progress closely related to the themes of this paper.

QUESTIONS
We conclude by discussing some potential future directions and open problems suggested by our work.In the special case of balanced rational normal scrolls, a stronger statement than Theorem 1.1 is proved in [CHV96, Theorem 3.8]: their blowup algebras have rational singularities if char(k) = 0 and F-rational singularities if char(k) > 0. These properties are stronger than Cohen-Macaulayness, and they also imply normality; it is therefore natural to ask whether they hold for all 2-determinantal ideals.Since these singularities are preserved by deformation, the machinery of our paper applies, and it suffices to study the blowup algebras of the most special ideal L. Question 9.1.Let L be the 2-determinantal ideal of KW type (1 d ; e).Are the blowup algebras R(L), F(L) normal?Do they have rational or F-rational singularities?
We point out that the main technique of [CHV96], namely degenerations to toric rings by means of Sagbi bases, cannot be applied.In fact, in general the natural generators of L do not form a Sagbi basis for any monomial order, and the same is true for non-balanced scrolls.
The most interesting next step in this area is perhaps the investigation of blowup algebras arising from matrices with more than two rows.For instance, very little is known about their singularities or degrees of defining relations; see [CP22] for some results in this direction.However, in principle, one may apply the same strategy developed in this paper: for matrices of arbitrary size, the corresponding determinantal ideals are still parametrized by well-behaved Hilbert schemes [KMR20], and the results of [BCV15] are still valid.The main goal would then be to identify the most special ideals in this parameter space, analogously to Theorem 4.3.Of particular interest would be the case of matrices of Hilbert-Burch size, that are sufficiently general in the sense of [BCV15, Theorem 3.7]: their determinantal ideals define arithmetically Cohen-Macaulay subschemes of codimension 2, and their Hilbert schemes are among the best behaved and most studied in the literature.
Problem 9.2.Fix m, n ∈ N. Let S = Sym(k n+1 ) and let p(z) be the Hilbert polynomial of S/I m (M) where M is an m × (m + 1) matrix of linear forms with codim I m (M) = 2. Let H ⊆ Hilb p(z) (P n ) be the set parametrizing ideals I m (M) such that codim I 1 (M) = n+ 1 and codim I j (M) min(n+ 1, m+ 1− j) for 2 j m − 1. Find a (small) set of ideals L ∈ H such that every I ∈ H degenerates to one of them.
Another interesting question is what happens to the blowup algebras when we drop the assumption that I m (M) has the expected codimension.This question is probably hopeless for matrices of arbitrary size, but it might be reasonable in the case of matrices with 2 rows, where Kronecker-Weierstrass normal forms are still available.An interesting case is that of Kronecker-Weierstrass normal forms without nilpotent blocks, but with possibly repeated eigenvalues in the Jordan blocks.Computational evidence suggests that the defining relations may still be similar to those found in Section 5. We conclude with a combinatorial consideration.For each c, d there are two different known squarefree degenerations of the special fiber ring F(I) of some I ∈ H c,d : one was established in [CHV96], while the other follows from Theorem 6.1.It would be interesting to compare the combinatorics and homology of the two simplicial complexes ∆ CHV and ∆ F determined by these two degenerations.For example, computational evidence suggests the following question about the graded Betti numbers of their Stanley-Reisner ideals: Question 9.4.Is β i,j I(∆ F ) β i,j I(∆ CHV ) for all i, j?
The question is already interesting in the case c − d 4, in which there are only Plucker relations, and F(I) is simply the coordinate ring of the Grassmannian variety of lines: then, ∆ CHV is the non-nesting complex, and ∆ F is the non-crossing complex.They have been studied e.g. in [PPS10,SSW17].In this case, ∆ F is Gorenstein, while ∆ CHV is not, cf.

Definition 4. 1 .
Let c, d ∈ N and let n = c + d.We denote by H c,d ⊆ Hilb p(ζ) (P n ) the locus of 2determinantal ideals in S = Sym(k n+1 ) of codimension c that have d scroll blocks in their Kronecker-Weierstrass normal form.Equivalently, H c,d ⊆ Hilb p(ζ) (P n ) is the locus of 2-determinantal schemes that are not cones.It is clear that H c,d = ∅ if and only if 0 d c + 1.If c 2, the closure H c,d is a generically smooth irreducible component of Hilb p(ζ) (P n ), see for instance [KMR20, Theorem 3.3].The case c = 1 is not very interesting: we have d 2, Hilb p(ζ) (P d+1 ) is just the projective space of quadric hypersurfaces, and H 1,d is the locus of determinantal quadrics.Let L (λ;µ) ⊆ H c,d denote the locus of 2-determinantal ideals of KW type (λ; µ).By Proposition 3.1 and Definition 3.2, the subset L (λ;µ) is the image of a morphism U × Aut k (S) → Hilb p(ζ) (P n ), where U is the affine variety parametrizing tuples of distinct eigenvalues and Aut k (S) is the group of k-algebra Theorem 4.3.For every ideal I ∈ H c,d , there is a sequence of degenerations to an ideal of KW type (1 d ; c − d + 1).Moreover, the locus L (1 d ;c−d+1) is the unique closed stratum of H c,d , and it is a single Aut k (S)-orbit.
Theorem 4.3.Applying Lemmas 4.4 and 4.5 repeatedly, it follows that for any ideal I ∈ H c,d there is a sequence of degenerations to an ideal of KW type (1 d ; c − d + 1).Consider the action of Aut k (S) on Hilb p(ζ) (P n ).It restricts to an action on H c,d .It follows by Lemma 3.4 that L (1 d ;c−d+1) is a single orbit of Aut k (S).By Lemmas 4.4 and 4.5, all orbits different from L (1 d ;c−d+1) are not closed in H c,d .Since orbits of minimal dimension are closed, it follows that L (1 d ;c−d+1) is closed in H c,d .
Lemma 4.6.Let I ∈ H c,d .The Hilbert functions of R(I) and F(I) depend only on c, d.Proof.By [BCV15, Theorem 3.7], the graded Betti numbers of each power I k are uniquely determined by c, d, k.It follows that the Hilbert functions of R(I) and F(I) are uniquely determined by c, d.

Notation 5. 1 .
Let d, e ∈ N be non-negative integers such that d + e 3, and let c = d + e − 1.Consider the following matrix and determinantal ideal

2
denote the vertex set of ∆ F .We identify vertices with open intervals of the real line R, and use concepts such as inclusion, length, and intersection.Translating the two classes of generators of I(∆ F ), a subset F ⊆ V is a face of ∆ F if and only if two conditions hold:( †) Any two intervals in F are non-crossing: either they are disjoint or one is contained in the other.( ‡) F contains no two intervals that are disjoint and contained in (d + 2, c + 1).
Proof of Theorem 1.1.Let S = Sym(k n+1 ) be a polynomial ring over a field k, and let I ⊆ S be the ideal of minors of a 2 × (c + 1) matrix M of linear forms with codim(I) = c.If c = 1, then I is a principal ideal, thus, R(I) and F(I) are polynomial rings, and the conclusions of Theorem 1.1 are straightforward.Therefore, we may assume c 2. The formation of the blowup algebras and the Koszul and Cohen-Macaulay properties are compatible with extensions of the base field and with the adjunction of new variables.Therefore, we may assume k = k and codim I 1 (M) = n + 1.By Proposition 3.3, there are two possibilities, corresponding to having c n or c = n + 1.If c = n + 1, then it follows by Proposition 3.3 that I is the square of the maximal ideal of S. The conclusions of Theorem 1.1 are well-known in this case, since F(I) and R(I) are particularly simple toric rings.For instance, it follows by [D99, Theorem 2.6] and [HHV05, Theorem 5.1] that they are defined by squarefree quadratic Gröbner bases, in particular, that they are Koszul algebras, while the Cohen-Macaulay property follows by [BC17, Theorem 5.4].If c n, then I ∈ H c,d for d = n − c.Assume first that d > 0.

Question 9. 3 .
Let I = I 2 (M)where M is a concatenation of scroll blocks and Jordan blocks.Are the blowup algebras R(I) and F(I) defined by quadratic relations?