A characteristic free approach to secant varieties of triple Segre products

The goal of this short note is to study the secant varieties of the Segre embedding of the product P1 × Pa−1 × Pb−1 by means of the standard tools of combinatorial commutative algebra. We reprove and extend to arbitrary characteristic the results of Landsberg and Weyman [5] regarding the defining ideal and the Cohen–Macaulay property of the secant varieties. Furthermore we compute their degrees and give a bound for their Castelnuovo–Mumford regularities, which are sharp in many cases.


Introduction
The topic of this paper is the secant varieties of the triple Segre product of P 1 × P a−1 × P b−1 . For basic facts about tensor decomposition and Cohen-Macaulayness of secant varieties of Segre products we refer the reader to Oeding's paper [7] or to the book of Landsberg [4]. We will work with 3-tensors of size (2, a, b) with 2 a b. The variety of rank-1 tensors is denoted by Seg (2, a, b) and corresponds to the image of the Segre embedding of P 1 × P a−1 × P b−1 in P 2ab−1 . We will further denote by σ t (2, a, b) the t-secant variety of Seg (2, a, b), that is, the variety of rank-t tensors. The defining ideal of Seg (2, a, b) in R = K[x ijk : (i, j, k) ∈ [2] × [a] × [b]] will be denoted by I(a, b). The ideal defining the secant variety σ t (2, a, b) will be denoted by I(a, b) {t} .
We recall that Seg(2, a, b) is a well understood toric variety whose defining ideal is the Hibi ideal of [ The unfolding (a.k.a. flattening) is a transformation that reorganizes a tensor into a matrix. For a tensor of size n 1 × n 2 × · · · × n d , the k-th unfolding is a matrix of size n k ×(n 1 · · · n k−1 n k+1 · · · n d ). The rows are indexed by the k-th index, and the columns are indexed by the vectors of the remaining indices. The order on the row and column indices is not important. In the following we will order them lexicographically.
. We remark that for X of size 2 × a × b the second and the third unfolding of X are the block matrices They can be combined in the single arrangement: The (t + 1)-minors of the various unfolding matrices are contained in the ideal defining the t-secant variety of the Segre variety but, in general, one needs extra generators. On the other hand we will see that for the tensor of size (2, a, b) the (t + 1)-minors coming from the unfolding matrices are enough to generate I(a, b) {t} .
In the following for a matrix M with entries in a ring R and t ∈ N we will denote by I t (M ) the ideal of R generated by all the t-minors of M .

Gröbner bases and secant ideals
In this section we quickly recall the results of Sturmfels and Sullivant [9] and of Simis and Ulrich [8] that we will use.
We denote by I {r} the r-fold join of I with itself, that is, If K is algebraically closed and I defines an irreducible variety X ⊂ P N , then I {r} is the defining ideal of the rth secant variety X {r} of X, defined as: Moreover the r-fold join of an initial ideal contains the initial ideal of r-fold join ideal. If the equality holds for every r then the term order is said to be delightful for the ideal I.

The ideal of two minors and the associated simplicial complex
Let K be a field and let X = (x ijk ) be the tensor of indeterminates of size 2 × a × b, with 2 a b.
Let R = K[x 111 , x 112 , . . . , x 2ab ] be the polynomial ring with indeterminates the entries of X . From now on we denote by τ any diagonal term order, that is, a term order such that the initial term of every minor of X {2} and of X {3} is its diagonal term. For example the lexicographic order induced by One can easily check that in this case the defining ideal of the Segre embedding of P 1 ×P a−1 ×P b−1 in P 2ab−1 , that is, the Hibi ideal associated to the poset [2]×[a]×[b], is indeed the ideal generated by the 2-minors of the second and third unfolding, that is, For example, the Hibi relation x 213 x 124 − x 113 x 224 is not a minor in X {2} or X {3} but can be written as a sum of a 2-minor of X {2} and a 2-minor of X {3} : We start by identifying a Gröbner basis for the ideal I(a, b) itself. Proof. It is not restrictive to consider the lexicographic order τ induced by x 111 > x 112 > x 113 > x 121 > · · · > x 1ab > x 211 > · · · > x 2ab . We first consider the following subsets of 2-minors in I 2 : It is well known and easy to check that G {2} and G {3} are Gröbner basis of the ideals they generate with respect to τ . Hence it is enough to consider the S-polynomials S(f, g) of f, g with f ∈ M {1} and g ∈ M {2} whose initial monomials in(S(f, g)) are not relatively prime. There are two cases.
To conclude we have to consider three more situations, and arguing as before.
Finally, if α 2 =α 2 and α 3 <α 3 , then in(S(f, g)) = x 1α2α3 x 1α2β3 x 2β2α3 and we have Thus in these three situations S(f, g) reduces to zero modulo M ; this finishes the proof.
As we have seen in Example 1.1 instead of looking at the second and the third unfolding of the tensor X separately, we can combine them into a single arrangement: ) is generated by the 2-minors in the arrangement and the initial ideal in (I(a, b)) is generated by the 2-diagonals in the arrangement. We introduce a partial order in the set of variables so that the 2-diagonals are exactly the pairs of comparable elements. To this end, we identify the set of the variables with: In P we introduce the following partial order: With this notation, the generators of in (I(a, b)) are exactly the pairs of distinct comparable elements in P . Now, in (I(a, b)), being a square-free monomial ideal, corresponds to a simplicial complex that we denote by ∆ 2 . Here we use the index 2 to recall the generators of the associated ideal are the 2-diagonals, that is, the pairs of comparable elements in P .
Recall that an antichain in the partially ordered set P is a set of elements no two of which are comparable to each other. Therefore the elements of ∆ 2 are the antichains of P . Moreover a saturated (or maximal) antichain of P is an antichain that is, maximal with respect to inclusions. Therefore the saturated antichains of P are the facets of P . A We will represent P with the matrix orientation, that is, with (1,1) in the top left corner and (a, 2b) in the bottom right corner. With this representation a path consists of a sequence of steps to the left and steps down. Proof. Let F be a path in P starting in (1, b + h) and ending in (a, h), with 1 h b, and passing through (k, b + 1) and (k, b). We first prove that F is a facet. It is clear that no pair of points in F is comparable, thus F ∈ ∆ 2 . Since we know that ∆ 2 is the simplicial complex associated to in (I(a, b)) and R/I(a, b) has dimension a + b, it follows that the facets of ∆ 2 have at most cardinality a + b. Hence F is a facet of ∆ 2 .
To conclude we show that every facet in ∆ 2 is a subset of a path F as described above. Let E ⊂ P be in ∆ 2 , that is, E does not contain any pair of comparable elements. Note that by the first condition of we have that E is an antichain, so it is of the form with i 1 · · · i α h 1 · · · h β and j 1 · · · j α b < b + 1 k 1 · · · k β . Moreover we can assume that j t+1 > j t whenever i t+1 = j t , and k t+1 > k t whenever h t+1 = h t . We prove that E is contained in a path F starting in (1, b + j 1 ) and ending in (a, j 1 ).
Remark 3.3. Let F be a path in P , starting in (1, b + h) and ending in (a, h), with 1 h b, and passing through (k, b + 1) and (k, b). It is clear that F corresponds to a path F in P = {(i, j) : 1 i a, 1 j 2b or a + 1 i 2a, 1 j b} starting in (1, b+h) and ending in (a+k, 1), with k a, and passing through (k, b+1), (k, b), and (a, h) (see Figure 1). Thus one can also rephrase the theorem in term of paths in P . In the following we will use both the descriptions of ∆ 2 , as subset of P and of P .

Secant ideals
Our goal is to prove:  To simplify the notation in the following we will denote by I t+1 the ideal I t+1 (X {2} ) + I t+1 (X {3} ).
The ideal I(a, b) defines Seg(2, a, b) and is generated by the 2-minors of the arrangement W , that is, I 2 = I(a, b). Hence the secant variety σ t (2, a, b) is contained in the variety of tensors whose second and third unfolding are of rank at most t. Hence it follows that: consists of the subsets of P that can be decomposed into the union of t antichains. According to Mirsky's theorem, they are exactly the subsets of P that do not contains chains of t + 1 elements. Equivalently the ideal of ∆ is generated by the chains of t + 1 elements, that is, the leading terms of the (t + 1)-minors of W .

Secant ideals
Our goal is to prove:  To simplify the notation in the following we will denote by I t+1 the ideal The ideal I(a, b) defines Seg(2, a, b) and is generated by the 2-minors of the arrangement W , that is, I 2 = I(a, b). Hence the secant variety σ t (2, a, b) is contained in the variety of tensors whose second and third unfolding are of rank at most t. Hence it follows that: consists of the subsets of P that can be decomposed into the union of t antichains. According to Mirsky's theorem, they are exactly the subsets of P that do not contains chains of t + 1 elements. Equivalently the ideal of ∆ is generated by the chains of t + 1 elements, that is, the leading terms of the (t + 1)-minors of W .
Proof of Theorem 4.1 parts (1) and (2). Denote by J t the ideal generated by the initial terms of the (t + 1)-minors of the arrangement W , that is, the monomials correponding to the t + 1-diagonals of P . Notice that we have the following relations To prove 4.1 part (3) we need to describe better the facets of the simplicial complex ∆ t+1 associated to in (I(a, b) {t} ). As a special case of Theorem 4.1 (2) we have that if t min (2a, b) the ideal I(a, b) {t} is trivial and if a t < min(2a, b) then I(a, b) {t} is a generic determinantal ideal and hence well understood. So in the remaining part of the paper we will assume that t < a.
We recall this well-known result. Denote by F(∆) the set of the facets of a simplicial complex ∆.
Secondly F 2 is the "right most" path in P from (1, 2b − 1) to (a, b − 1) that does not intersect F 1 and for i = 3, . . . , t, F i is the "right most" path in P from (1, 2b − i + 1) to (a, b − i + 1) to that does not intersect F i−1 . For example the picture 2 shows this construction in the extremal case t = a = 6, b = 10. In the following we say that an element (x, y) of a path G in P is a low-right corner of G if also (x − 1, y) and (x, y − 1) are in G. By using the description of the facets of ∆ t+1 given above we prove the following.
First we define a partial order on the set of the facets of ∆ t+1 . Let F = G 1 ∪· · ·∪G t and F = G 1 ∪ · · · ∪ G t be two facets of ∆ t+1 , where G i is a path starting in (1, b + h i ) and ending in (a, h i ) and G i is a path starting in (1, b + k i ) and ending in (a, k i ). Assume that h 1 < · · · < h t and k 1 < · · · < k t . Set Extend to a total order on the set of the facets of ∆ t+1 . To prove that ∆ t+1 is shellable, we need to show that, given any two facets F < F , there exists x ∈ F F and a facet F such that F < F and F F = {x}.
Let F < F be two given facets decomposed as above. Since F F , we may consider the least integer i such that G i ⊆R Gi , with i t. Thus there exists y ∈ G i such that y ∈R Gi , that is, y ∈ L Gi . Choose such a y with the smallest row index possible, and note that R y ∩ G i = ∅. Now we consider two cases, according to the existence of a low-right corner x of G i which is in R y , cf. Figure 3.
Case 1. Assume that such a corner does not exist. Let y = (r, s); note that since y ∈ L Gi , then there exist x ∈ G i such that y ∈ L x , and by assumption x = (r , k i ).
, and there would be a low-right corner of G i belonging to R y , a contradiction.
, and there exists an element y ∈ G i+1 such that y ∈R Gi+1 . So we can start again arguing on y as we did on y, by using Case 1 or Case 2.
Case 2. Suppose that such a corner exists. Since y ∈ G i and the path of F are disjoint, one has that y ∈ R G i−1 . By the minimality of i, G i−1 ⊆R Gi−1 , thus y ∈ R Gi−1 . Since y ∈ L x , it follows that x L ∈ R Gi−1 and, consequently, {x} is a facet of ∆ t such that F F , thus F < F , as desired.
It remains to consider the case in which x ∈ F , that is, i < t and x ∈ G i+1 . Since x ∈R Gi+1 , we can start again applying to x the arguments we have used for y, following Case 1 or Case 2.
It is clear that this procedure concludes after a finite number of steps, proving that ∆ t+1 is shellable.
As a consequence we can conclude the proof of Theorem 4.1.  Proof. Since ∆ t+1 is pure, the multiplicity of R/I(a, b) {t} is equal to the number of the facets of ∆ t+1 . The set of the facets of ∆ t+1 can be written as the disjoint union of the set of non-intersecting paths with starting points By a well known formula of Gessel-Viennot [2], the number of non-intersecting paths with starting points Q 1 , . . . , Q t and ending points P 1 , . . . , P t is the determinant of the matrix whose (i, j)-th entry is the number of paths from Q i to P j which is easily seen to be equal to This gives the formula for the multiplicity. Given a shellable simplical complex ∆ with a shelling G 1 , . . . , G v we set r(G i ) = {x ∈ G i : there exists j < i such that G i G j = {x}}.  {x ∈ F i : x is a right turn of F i or x is the starting point of F i }.
Since any path F i has at most a − 1 such turns, we have that |r(G)| at and hence reg(R/I(a, b) {t} ) at. If b 2t then one can actually find a facet G of ∆ t+1 such that |r(G)| = at proving that for b 2t one has reg(R/I(a, b) {t} ) = at. The construction of the facet G such that |r(G)| = at is illustrated by the following example, where Algebraic Combinatorics, Vol. 3 #5 (2020) (a, b) = (5,8), t = 4, and a facet G is represented with a dot in every point belonging to r(G): In the case a = b = 2t the facet with |r(G)| = at described in the proof is indeed the only facet of ∆ t+1 with that property, as in this example, with (a, b) = (6, 6), t = 3: This implies that the highest non-zero entry of the h-vector of ∆ t+1 is 1. The latter is a necessary condition for the R/I(a, b) {t} to be Gorenstein. This leads to the following conjecture:  1 + 36x + 666x 2 + 8436x 3 + 68526x 4 + 366660x 5 + 1330644x 6 + 3296124x 7 + 5650866x 8 + 6762316x 9 + 5650866x 10 + 3296124x 11 + 1330644x 12 + 366660x 13 + 68526x 14 + 8436x 15 + 666x 16 + 36x 17 + x 18 .