A note on secant defective varieties and Clifford modules

We generalise a construction of Landsberg, which associates certain Clifford algebra representations to Severi varieties. We thus obtain a new proof of Russo's Divisibility Property for LQEL varieties.


Introduction
The geometry of secant-defective varieties is surprisingly rich.In the early 20 th Century, the subject captured the attention of several members of the Italian School of Algebraic Geometry and important results appear in numerous beautiful old papers, such as those of Scorza [15], Severi [16], and Terracini [17].
In the 1980s the subject enjoyed a renaissance, largely due to a series of breakthroughs made by Zak [18].Zak's solution of Hartshorne's linear normality conjecture [3] lead to his classification of maximally-secant-deficient varieties, which he named Severi varieties.He showed that there are exactly four Severi vareties, that they correspond to normed division algebras, and that their dimensions are 2, 4, 8, 16.The hardest part of the classification was establishing the dimension restriction.
In 1996, an intruiging paper of Landsberg [8] appeared in which he showed that the extrinsic geometry of a Severi variety induces certain Clifford algebra representations.Using the classification of Clifford modules, it is then trivial to see that the dimensions of the Severi varieties must take the values already established by Zak.The main purpose of this note is to show that Landsberg's Clifford modules may be generalised.Severi varieties are examples of a class of varieties introduced by Russo [13] in 2009, called LQEL varieties and we show that Landsberg's construction works in this more general setting.
Actually the generalisation is only an extremely mild extension of Landberg's results.However we believe it is worth highlighting because, together with the classification of Clifford modules, it provides a new proof of Russo's Divisibility Property for LQEL varieties (corollary 2.8).

Secant defective varieties and Clifford modules
We shall follow Landsberg [6,8] closely and so recall his constructions and notation1 .Our primary object of concern is a subvariety of projective space.We write: to indicate that the variety X is n-dimensional and that the embedding has codimension a.We work over C throughout, and assume that X is smooth, irreducible, and non-degenerate2 , with secant deficiency δ ≥ 1.

Second fundamental form
We recall [2] that the second fundamental form of an embedding X ⊆ P n+a is a section of Hom(S 2 T X, N), where S 2 T X is the symmetric square of the tangent bundle and N is the normal bundle.Thus for x ∈ X, we have a symmetric bilinear map: When we have chosen a point x ∈ X and there is no possibility of confusion, we will write T for T x X, N for N x and II for II x .Taking the transpose, we also regard the second fundamental form as a linear system of quadrics: A key observation is that global properties of X are visible infinitesimally via the second fundamental form, and exceptional global properties tend to produce linear systems of quadrics with exceptional properties.
For example, if X has a smooth dual variety, then at a general point II * is a linear system of quadrics of constant rank, and from this follows Landman's parity theorem for dual-deficient varieties (see [6,Theorem 12.4.8 and Corollary 12.4.10]).
We shall show that if X has the exceptional property that the Gauss map of a general tangential projection has zero-dimensional fibres, then its second fundamental form can be used to construct certain Clifford modules, and from this follows Russo's Divisibility Property for secant-deficient varieties (see [13,Theorem 2.8] or [14, Theorem 4.2.8]).

Tangential projections
We now assume X is non-linear3 make two definitions to fix notation: Definition 2.1.Let x ∈ X and let T x X ⊆ P n+a be the embedded tangent space at x. Away from T x X we define a rational map: where N is the fibre of normal bundle of X at x.The map π x is known as the tangential projection map at x.
Definition 2.2.Let x ∈ X and II be the second fundamental form at x. Away from Baseloc II * we define the rational map: We recall that the closures of the images of π x and ii coincide and have dimension n − δ (see e.g., [14, Proposition 2.3.5] and its proof).Let this (n − δ)-dimensional irreducible subvariety be: and let: Sec(X) ⊆ P n+a , be the (2n + 1 − δ)-dimensional secant variety of X, then we note for future reference that: codim Z = codim Sec(X). (1)

Second fundamental form of a tangential projection
The following is essentially a restatement of [8,Lemma 6.6].Given a general point x ∈ X and a general4 vector v ∈ T , let z = ii([v]).It follows from the definition of ii that there is a natural commutative diagram: and II v is the map: We thus have natural exact sequences: and: where N z Z is the normal bundle of Z ⊆ PN at z.The maps ρ T and ρ N fit into the following commutative diagram: where ĨI is the second fundamental form of Z at z.

Singular locus of the second fundamental form
Griffiths and Harris noticed that the quadrics of the second fundamental form are all singular along the fibres of the Gauss map.In fact they proved [2, (2.6)]: where F is the fibre of the Gauss map through x and for any A ⊆ N * , Singloc(A) is the intersection of all the singular loci: A key insight of Landsberg [8] was that there are natural subsystems A ⊆ N * for which Singloc(A) captures more delicate geometric features of X. Indeed it follows from (3) that there is a natural exact sequence: and so for v ∈ T , Landsberg defined: and studied the middle term Singloc(Ann(v)) appearing in (5).

Clifford modules
The simplest class of secant-deficient varieties are those for which the Gauss map of a general tangential projection has zero-dimensional fibres.For emphasis we state a key consequence of this property: Lemma 2.3.Let X ⊆ P n+a be a smooth, irreducible, non-degenerate variety of secant deficiency δ ≥ 1.Let x ∈ X, v ∈ T x X be general and let Z ⊆ PN be the closure of the tangential projection at x. Then the Gauss map of Z has zero-dimensional fibres if and only if: • the quadratic Veronese embeddings ν 2 (P n ) ⊆ P n(n+3) 2 for n ≥ 2 (δ = 1), • the binary Segre embeddings P n × P m ⊆ P mn+m+n for m + n ≥ 3 (δ = 2), • the rank-2 Plücker embeddings G(2, n) ⊆ P (n−2)(n+1) 2 for n ≥ 5 (δ = 4), • the 16-dimensional Severi variety in P 26 (δ = 8), as well as their linear projections.We recommend [14] for further discussion.
Remark 2.5.In [14, Definition 3.3.1],given general points x, y ∈ X and general p ∈ Sec(X) on the line xy, Russo defines the contact locus Γ p ⊆ X as: and notes that by Terracini's lemma: where Σ p is the entry locus of X with respect to p ∈ Sec(X).Let Z = π x (X) be the closure of the tangential projection at x and F ⊆ Z be the fibre of the Gauss map of Z through z = π x (y).As argued by Russo in the proof of [14, Lemma 3.3.2]the irreducible components of π −1 x (F ) and Γ p through y coincide.We should thus have a natural exact sequence of tangent spaces: The line yp naturally determines a vector v ∈ T y X, and we expect: so that (8) can be interpreted as (5).Given this, [14, Lemma 3.3.2(2)] would provide an alternate proof of lemma 2.3 above.
We come at last to our main point: Theorem 2.6.Let X ⊆ P n+a be an smooth, irreducible, non-degenerate variety of secant deficiency δ ≥ 1 such that Sec(X) ≠ P n+a .Suppose that the Gauss map of the tangential projection at a general point x has zero-dimensional fibres, and let v ∈ T x X be general.Then T Singloc(Ann(v)) carries a natural Clifford module structure over ker II v .
Proof.Let Z = π x (X) ⊆ PN be the closure of the image of the tangential projection at x.The result we need is exactly [8,Lemma 6.26] except that we have made no assumption about Z being a cone (instead assuming that its Gauss map has zero-dimensional fibres) and we do not assume that Z is a hypersurface.In view of (1), Z is a hypersurface if and only if Sec(X) is.However since the linear projection from a linear subspace which does not meet Sec(X) is an isomorphism, we may select a maximal such subspace and project down to the case that Sec(X) is a hypersurface; the lemma then applies, and our proof is complete.
For the benefit of readers who wish to compare with [6], we provide a reference for the argument as presented there.The key equation is [6, (12.22) page 374]: The key assumption required for the derivation is S = 0 where: which follows from lemma 2.3.
Remark 2.7.We can restate theorem 2.6 without referring to the second fundamental form as follows.
Let X be as in theorem 2.6 and let p ∈ Sec(X) and x ∈ X be general points.Let Q ⊆ X be the irreducible component of the p-entry locus through x and let Q ′ ⊆ Q be the corresponding 5 irreducible component of the tangent locus through x.Then T x Q ′ carries a non-degenerate quadratic form and the fibre N x Q X of the normal bundle of Q in X is a Clifford module for the Clifford algebra Cl(T x Q ′ ).
We emphasise the following corollary: Corollary 2.8.Let X be as in theorem 2.6 then: Proof.The result follows immediately from theorem 2.6 together with the classification of Clifford modules.Indeed if there exists a k-dimensional Clifford module of a non-degenerate l-dimensional complex quadratic form, then: where p = 2 ⌊ l 2 ⌋ .The reason is that the Clifford algebra of the quadratic form is the matrix algebra C p×p if l is even or C p×p ⊕ C p×p if l is odd (see e.g., [1,Table 1]) and the natural action of C p×p on C p is its unique irreducible representation.
Remark 2.9.The divisibility condition established in corollary 2.8 was first proved by Russo and appeared in [13,Theorem 2.8] (see also [14,Theorem 4.2.8]).The proof involved an inductive study of the Hilbert scheme of lines through a general point of an LQEL variety.
A second proof (due to the author) appeared as [11,Corollary 2.6].This proof was topological and the key was a calculation in topological K-theory.
We now have a third proof (albeit with slightly different assumptions) and this time it is Clifford module theory that is the key.
It would be interesting to explore the relationship between the second and third proofs given the deep connections between K-theory and Clifford modules identified by Atiyah, Bott, and Shapiro in [1].The first step should be to argue that Landsberg's construction actually provides bundles of representations of Clifford algebras, as x varies over a general tangent locus.
Remark 2.10.Note that the proof of corollary 2.8 shows that the 2 which appears in the expression (δ − 1) 2 corresponds to the mod-2 periodicity of Morita equivalence classes of complex Clifford algebras.Thus it is the same 2 which appears in complex Bott periodicity.
Remark 2.11.A quite different connection between Clifford modules and Severi varieties arises in the context of 'Clifford structures', introduced by Moroianu and Semmelmann in [10].The Severi varieties appear in the classification of parallel non-flat even Clifford structures in [10] (see also [12]).It might be interesting to explore whether these Clifford structures have any relationship to Landsberg's Clifford modules.

A remark about the δ ≤ 8 problem
Let X ⊆ P n+a be a smooth, irreducible, non-degenerate, subvariety with Sec(X) ≠ P n+a .It follows from Zak's proof of Hartshorne's linear normality conjecture that the secant deficiency satisfies: In the course of their exposition [9] of Zak's work, Lazarsfeld and Van de Ven highlighted that all known examples of X as above satisfy δ ≤ 8.They thus posed the problem to investigate whether δ could be arbitrarily large (see [9, §1f, page 19]).In view of ( 9), any X with δ > 8 must have dimension n ≥ 18.
Very little progress has been made on this problem in the 40 years since it was first posed.Kaji [7]  We mention this problem here, to highlight that by combining known results, one may sharpen (9) slightly as follows: Lemma 3.1.Let X ⊆ P n+a be a smooth, irreducible, non-degenerate subvariety with Sec(X) ≠ P n+a .Suppose n ≥ 17, then: Proof.For a general tangential projection of X, let γ be the dimension of the fibre of its Gauss map and ξ its dual deficiency.Suppose first that γ = 0. We may assume δ ≥ 1 (otherwise there is nothing to prove) so by Scorza's Lemma [14, Theorem 3.
as required 6 .
Note that lemma 3.1 increases the dimension restriction on a variety with δ > 8 slightly to n ≥ 19.One might hope to make further progress by studying varieties for which γ = 1 and then arguing as in lemma 3.1 but with three cases corresponding to whether γ is 0, 1, or at least 2.
has shown that any variety with δ > 8 must be non-homogeneous but otherwise the problem remains completely open: all known examples still satisfy δ ≤ 8 and the 16-dimensional Severi variety remains the only variety known to achieve δ = 8.The problem remains open even for the very special class of LQEL varieties (see [14, chapter 4.4, page 113] as well as [4, Remark 3.8] and [5, Conjecture 4.5]).

)
Proof.Applying Griffiths and Harris's result[2, (2.6)] to Z, we know that the Gauss map of Z has zero-dimensional fibres if and only if the third term in (5) vanishes.Bearing in mind (2), the conclusion is clear.