Linear subspaces of symmetric tensors whose non-zero elements have at least a prescribed rank

We introduce large vector spaces M of multivariate homogeneous polynomials with a prescribed lower bound for the rank of each non-zero element of M.

In the set-up of Theorem 1.1, the set S is the only set evincing the rank of P (Proposition 2.4), i.e., r X m,d (P) = r and S is the only set A ⊂ P m with cardinality r such that P ∈ ν d (A) . See Proposition 2.2 for a stronger statement if m = 2.
We recall that a finite set S ⊂ P s is said to be in linearly general position if dim( E ) = min{s, (E) − 1} for all E ⊆ S. When m = 2, Theorem 1.1 is quite good (Remark 2.3), but when m d it says almost nothing. For any m ≥ 2, we prove the following result. We work over an algebraically closed base field K (see Remarks 2.5 and 3.5 for more general fields, Remark 3.6 for a discussion of the positive characteristic case).
We thank the referees whose advices improved the exposition. For each integer t ≥ 1, the t-secant variety σ t (X m,d ) ⊆ P n of X m,d is the closure inside P n of all linear spaces A with A ⊂ X m,d and (A) = t. The border rank b X m,d (P) of P ∈ P m is the first integer t such that P ∈ σ t (X m,d ). When b X m,d (P) ≤ d + 1 there is a zero-dimensional scheme Z ⊂ P m such that deg(Z ) = b X m,d (P) and P ∈ ν d (Z ) ([9], Proposition 11, [12], Lemma 2.16). We say that any such Z evinces the border rank of P.
We first do the case m = 2, because in this case [16] is a very powerful tool (which is also stated and proved in arbitrary characteristic).  [16], Corollaire 3, and get that either τ = s − 3 + z/s and A ∪ W is the complete intersection of a curve of degree s and a curve of degree z/s or there is an integer t ∈ {1, . . . , s − 1} and a curve T ⊂ P 2 such that deg(T ) = t and deg(T

Proposition 2.2 Fix integers d, k such that d
(ii.1) First assume τ = s − 3 + z/s and that A ∪ W is the complete intersection of a curve of degree s and a curve of degree z/s. In particular, W red is contained in a curve of degree s. Since W red has general postulation, we get w ≤ ( We saw before step (i) that deg(L ∩ W ) ≤ 4 and that if equality holds, then L = L i, j for some i, j. If r ≤ d − 3, we get deg(W ∩ L) ≥ 5, a contradiction, concluding the proof of part (a). If r = d − 2, we get L = L i, j for some i, j and A ⊂ L i, j . Since P ∈ ν d (A) , we get P ∈ ν d (L i, j ) , concluding the proof of part (b). (iii) Take the set-up of part (c). By concision (either [11] [1,2,10,14].
By the Claim there is a minimal subscheme W 1 ⊆ W and a minimal subscheme Moreover, (W 1 ) red is general in P 2 . We are in the set-up of part (a) of Proposition 2.2 and we adapt step (ii) of its proof. Let τ be the maximal integer t such that h 1 (P 2 , I

Proofs of Theorem 1.2 and Corollary 1.3
For any zero-dimensional scheme Z ⊂ P m and any hyperplane H ⊂ P m , the residual scheme Res H (Z ) of Z with respect to H is the closed subscheme of P m with I Z : I H as its ideal sheaf. We have Res H (Z ) ⊆ Z and We need the following obvious lemma whose proof is omitted. The following two elementary lemmas are very classical and in characteristic zero stronger results are known (e.g. [8], Lemma 1.8). However, the statements and proofs must be characteristic free to hope any application to codes over a finite field.
then let S i be the set of all hyperplanes of P m . If A i = ∅, then let S i be the set of all hyperplanes of P m containing at least one point of A i . Every zero-dimensional scheme E ⊂ P m of degree ≤ m is contained in a hyperplane. Hence, if a i ≤ m − 1, For all i ≥ 0, we have the following residual exact sequences Since h 1 (I B 0 (d)) > 0, the exact sequences (3) for i ≥ 0 give the existence of a minimal integer e ≥ 0 such that Remark 3.6 Assume K algebraically closed with characteristic p > 0. We fix the degree d of the homogeneous polynomial we are interested in and hence we fix ν d . Fix any integer m > 0. We take as the definition of rank of P ∈ P n , n := m+d m − 1, the X m,d -rank, i.e., the rank with respect to the Veronese variety X m,d = ν d (P m ). If p > d (but only if p > d), we may translate this definition for a homogenous polynomial f = 0 as the minimal number of summands of d-powers of linear forms needed to obtain f . With our definition in terms of X m,d -rank if p > d, then the case m = 1 is true ( [20], Theorem 1.44), but the case p ≤ d fails (but it fails in a controlled way ( [3]); for instance if p = d = 2, there is a unique point of P 2 (the strange point of the smooth conic X 1,2 ⊂ P 3 ) with X 1,2 -rank 3). In the set-up of Proposition 2.2 and in many other places, one can substitute the characteristic zero quotations of concisions with an explicit proof of that particular case using [11], Lemma 1, and then play as in the proof of Theorem 1.2; in the plane this game should be substituted with [16], as we did in step (ii) of the proof of Proposition 2.2. If p > d, part (c) of Proposition 2.2 is true. If p ≤ d instead of part (c) of Proposition 2.2, one can give the following statement whose proof follows from [7], Lemma 5.1, or [16].
As in part (c) of Proposition 2.2 take d ≥ 7 and P ∈ ν d (L i, j ) with r X 2,d (P) ≥ d − 2. Then, b X 2,d (P) = b X 1,d (P), r X 2,d (P) = r X 1,d (P), every subscheme of P 2 evincing the border rank of P with respect to X 2,d is contained in L i, j and every subset of P 2 evincing the rank of P with respect to X 2,d is contained in L i, j .