X -ranks for embedded varieties and extensions of ﬁelds

Let X ⊂ P r be a projective embedded variety deﬁned over a ﬁeld K . Results relating maximum and generic X -rank of points of P r ( K ) and P r ( L ) are given, where L is a ﬁeld containing K . Some of these results are algebraically closed for K and L . In other results (e.g. on the cactus rank), L is a ﬁnite extension of K .


Introduction
In this paper we fix an extension of fields, say K ⊂ L, a projective variety X defined over K and an embedding of X into a projective space P r defined over K. Thus P r (K) ⊂ P r (L). For each a ∈ P r (K) there are several different notions of ranks with respect to X(K) and X(L). In Section 3 we consider the case in which K is not algebraically closed and L is a finite extension of K (see Theorem 3.1), and in the rest of the paper we consider the case in which both K and L are algebraically closed.
Fix algebraically closed fields K ⊂ L. Take F ∈ {K, L}. Let X ⊂ P r be an integral and non-degenerate variety defined over K. We also assume that the embedding of X in P r is defined over K and that X is non-degenerate, i.e. X(K) spans P k (K). Thus X(L) spans P r (L). Set n := dim X. For any scheme or algebraic subset Z ⊂ P r (K) (respectively, Z ⊂ P r (L)) defined over K (respectively, over L) let Z K ⊆ P r (K) (respectively, Z L ⊆ P r (L)) denote the linear span of Z over K (respectively, over L). Note that Z K L = Z L for any Z ⊆ P r (K). For all positive integers t let S(X(F ), t) denote the set of all subsets of X(F ) with cardinality t. The set S(X(F ), t) is an irreducible quasi-projective variety of dimension nt. For any o ∈ P r (F ) the X(F )-rank r X(F ) (o) of o is the minimal cardinality of a subset of X(F ) containing o in its linear span. For any positive integer t let S(X(F ), o, t) denote the set of all S ∈ S(X(F ), t) such that o ∈ S F and o / ∈ S F for any S S. Each set S(X(F ), o, t) is constructible by a theorem of Chevalley (see [8,Ex. II.3.18]). Note that r X(F ) (o) is the minimal integer t such that S ∈ S(X(F ), t) = ∅. Now assume o ∈ P r (K). Since K is algebraically closed, it is easy to check (and wellknown) that r X(L) (o) = r X(K) (o) and that S(X(L), o, t) is the constructible L-set associated to S(X(K), o, t) (see Remark 2.1 for more details). In particular S(X(L), o, t) and S(X(K), o, t) have the same number of irreducible components and the bijection between their irreducible components preserves the dimension of the components. In particular S(X(L), o, t) = S(X(K), o, t) if and only if either S(X(K), o, t) = ∅ or S(X(K), o, t) is finite. For any positive integer t let σ t (X(F )) ⊆ P r (F ) denote the closure in P n (F ) of the union of all S F , S ∈ S(X(F ), t). Each σ t (X(F )) is irreducible and σ t (X(L)) is the L-variety associated to the K-variety σ t (X(K)). The first integer a such that σ a (X(F )) = P r (F ) is the same for F = K and F = L. It is often call the generic X(K)-rank (respectively, generic X(L)), because it is the X(K)-rank (respectively, X(L)-rank) of a non-empty open subset of P r (K) (respectively, P r (L)). For any positive integer t let R(X(F ), t) denote the set of all o ∈ P r (F ) such that r X(F )) (o) = t. Each R(X(F ), t) is constructible (Lemma 2.1). See Remark 2.1 for the definition and construction of the L-associated set of any constructible subset of P r (K).
It is easy to prove the following result (its proof is given after Lemma 2.1).

Theorem 1.1.
For each positive integer t the constructible L-set R(X(L), t) is the L-set associated to R(X(K), t).
The maximum among all X(F )-rank is the largest integer a such that R(X(F ), a) = ∅. Thus Theorem 1.1 has the following corollary. Take any o ∈ P r (F ). The open X(F )-rank or X(F ) (o) of o is the minimal integer t > 0 such that for all closed sets T X(F ) there is S ∈ S(X(F ), t) such that o ∈ S F and S ∩ T = ∅ (see [1,9]). Obviously or X(F ) (o) ≥ r X(F ) (o), but very often the strict inequality holds. For instance, or We also prove the following results.    Proof. Since R(X(F ), 1) = X(F ), we may assume t > 1 and use induction on the integer t. Since R(X(F ), t)∩R(X(F ), x) = ∅ for all x < t, it is sufficient to prove that A := ∪ 1≤x≤t R(X(F ), x) is constructible. The set E is the image of S(X(F ), t) by the evaluation map.

When K is not algebraically closed
Let K be a field which is not algebraically closed. We fix an inclusion K ⊂ K. Let X ⊂ P r be an embedding (defined over K) of the integral projective variety X defined over K. We assume that X(K) is non-degenerate, but we do not assume that X(K) spans P r (K) (we allow the case X(K) = ∅). For each a ∈ K let deg(a) be the degree of the minimum polynomial of a over K, i.e. the dimension of the K-vector space K(a). We fix a system of homogeneous coordinates x 0 , . . . , x r of P r (K). For each a = (a 0 : · · · : a r ) ∈ P r (K) with, say, a i = 0 the degree deg 1 (a) of a is the maximum of all integers deg(a j /a i ), 0 ≤ j ≤ r, and let deg 2 (a) be the the degree of the extension K(a 0 /a 1 , . . . , a r /a i )  Take o ∈ P r (K) and fix i ∈ {1, 2, 3}. Set t := r X(K) (o). Let DR i (X, K, o) denote the minimum of all deg i (S) for some S ∈ S(X(K, t). We say that a = (a 0 : · · · : a r ) ∈ P r (K) is separable over K if all ratios a j /a i with a i = 0 are separable over K. Obviously if a i = 0 it is sufficient to test all a j /a i . If K is perfect, then every a ∈ P r (K) is separable over K. The field K is perfect if either K is a finite field or char(K) = 0. Example 3.1. Take r = 2, K = F q and a = (1 : u : v) with u ∈ F q 3 \ F q and v ∈ F q 2 \ F q . We have deg 1 (a) = 3 and deg 2 (a) = 6.
The fact that all finite extensions of a finite field are Galois extensions has the following byproduct.
Proof. Set x := deg 2 (o) and y := DR 2 (X, K, o). Write {S} = S(X(K), o, t). Consider X over F q x . Since o ∈ P r (F q x ) and S is the unique element of S(X(K), t) computing the X(K)-rank of o, S is invariant for the Galois group of the extension F q y /F q x . Thus y ≤ (#S)x. We have t = #S.
For any field E ⊇ K let ρ(X(E)) denote the maximal integer t such that any subset of X(F ) with cardinality t is linearly independent. Of course, if E ⊂ E , then X(E) ⊆ E and hence ρ(X(E )) ≤ ρ(X(E)). If E is algebraically closed, it is easy to check that ρ(X(E )) = ρ(X(E)) for any field E ⊃ E. Remark 3.1. Fix o ∈ P r (K) and assume 2r X(K) ≤ ρ(X(K)). Then we have #S X(K), o, r X(K) = 1.

Remark 3.2.
Let ν d : P n − → P r , r = n+d n −1, be the d-Veronese embedding of P n , i.e. the embedding induced The cohomology of a projective space easily gives that ρ(ν d (P n )(E)) = d + 1 for any field E. In particular we may apply Remark 3.1 to any o ∈ P r (K) such that The proof of Proposition 3.1 gives the following result. By Remark 3.2, Proposition 3.2 may be applied to the d-Veronese embedding of any projective space, but just in a very restricted range of ranks.
Other notions of ranks for homogeneous polynomials are the slice rank and the Schmidt rank (often called strength). The recent preprint [10] by Lempert and Ziegler proves stronger versions of all our attempts related to this notion over a non-algebraically closed field with characteristic 0. Remark 3.3. Take any field K such that char(K) = 0 and let X ⊂ P r , r = n+d n − 1, be the image of the the d-Veronese embedding of P n . Fix a ∈ X(K) and o ∈ P r (K).
If we do not search for a small degree extension of K on which it is defined all points (or the set) defining the X(K)rank, then we may get far better bounds. We recall that the cactus X(K)-rank of a ∈ P r (K) is the minimal degree of a zero-dimensional scheme Z ⊂ X(K) whose linear span contains a (see [2,3,5,7]). Fix a finite extension L of K such that a is defined over K. We call cactus L-rank the minimal degree of a zero-dimensional scheme Z ⊂ X(K) defined over L and whose linear span contains a. We call strong cactus L-rank the minimal degree of a zero-dimensional scheme Z ⊂ X(K) such that all connected components of Z are defined over L and the linear span of Z contains a. Obviously every connected Z defined over L may be use to test the strong cactus rank. Theorem 3.1. Assume char(K) = 0. Let L be any finite extension of K. Fix an integer d ≥ 3. Let X(K) ⊂ P r , r = n+d n − 1, be the image of the d-Veronese embedding of P n . If d = 2k + 1 is odd, set N := 2 n+k n . If d = 2k + 2 is even, set N := n + k n + n + k + 1 n .
Then every a ∈ P r (L) has strong cactus L-rank ≤ N .
Proof. Fix b ∈ X(L). The proof of [3,Theorem 3] gives the existence of a zero-dimensional scheme Z ⊂ X(K) defined over L, spanning a and with Z red = {b}. Since Z is connected, it gives an upper bound for the strict cactus L-rank of a.