ON THE DUALS OF SEGRE VARIETIES

Hajime KAJI

Instituto Nacional de Matemática Pura e Aplicada, 22460–320 Rio de Janeiro, Brasil

                               kaji@impa.br

Department of Mathematical Sciences, Waseda University, Tokyo 169-8555, Japan

                           kaji@mse.waseda.ac.jp

Abstract. The reflexivity, the (semi-)ordinariness, the dimension of dual varieties and the structure of Gauss maps are discussed for Segre varieties, where a Segre variety is the image of the product of two or more projective spaces under the Segre embedding. A generalization is given to a theorem of A. Hefez and A. Thorup on Segre varieties of two projective spaces. In particular, a new proof is given to a theorem of F. Knop, G. Menzel, I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky that states a necessary and sufficient condition for Segre varieties to have codimension one duals. On the other hand, a negative answer is given to a problem raised by S. Kleiman and R. Piene as follows: For a projective variety of dimension at least two, do the Gauss map and the natural projection from the conormal variety to the dual variety have the same inseparable degree?

0. Introduction

Segre varieties are one of the most fundamental examples of projective varieties. The purpose of this article is to study the Segre varieties from the view-point of projective geometry, where a Segre variety is the image of the product of two or more projective spaces under the Segre embedding. Precisely speaking, we discuss about four points, namely, the reflexivity, the (semi-)ordinariness, the dimension of dual varieties and the structure of Gauss maps for Segre varieties (see §1 for the terminology).

A. Hefez and A. Thorup [3] studied the first three points for Segre varieties of just two projective spaces. According to their result [3, Theorem 2], such varieties are all reflexive for any characteristic of the ground field, for instance. However, this does not hold for general case. Our main result gives a generalization of the result of Hefez and Thorup, as follows:

Theorem. Let X be the image of the Segre embedding of ∏_{i=1}^{r} ℙ^{n_{i}} over the ground field K of characteristic p ≥ 0 with dimension n = ∑_{i=1}^{r} n_{i}, and assume that n_{i} ≥ 1 for all i and r ≥ 2. Then we have:

(1) The dual variety X^{∗} is a hypersurface if and only if 2n_{i} ≤ n for all i.

Typeset by AMS-TEX

1991 Mathematics Subject Classification. Primary 14N05, Secondary 14M12.

Key words and phrases. Segre variety, reflexivity, dual variety, Gauss map, symmetric matrix.

—2001/08/23; version 1.05—

(2) If 2n_{i} > n, then (a) X is reflexive with codim X^{∗} = 1 + 2n_{i} − n, and (b) X is semi-ordinary if and only if 2n_{i} = n + 1.

(3) If 2n_{i} ≤ n for all i, then the following conditions are equivalent: (a) X is not reflexive (or equivalently, not ordinary).

(b) X is semi-ordinary.

(c) p = 2 and n is odd.

I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky [1] proved the statement (1) over the complex number field ℂ (see [1, Theorem 0.1]), which follows also from more general results by F. Knop and G. Menzel [9]. Our proof is quite different from the known ones and seems to be much simpler.

For the fourth point, we have

Proposition 1. The Gauss map γ of the Segre embedding of ∏_{i=1}^{r} ℙ^{n_{i}} with n_{i} ≥ 1 for all i and r ≥ 2 is always an embedding in any characteristic.

Thus it turns out that, combining Theorem (3) and Proposition 1, one obtains a negative answer to a problem raised by S. Kleiman and R. Piene [8] as follows:

Problem (Kleiman-Piene [8, pp. 108–109]). For a projective variety X ⊆ ℙ^{N} of dimension n ≥ 2, let γ : X −−→ 𝔾(n, ℙ^{N}) be the Gauss map, CX the conormal variety, X^{∗} the dual variety, and π : CX → X^{∗} the natural projection. Then do γ and π have the same inseparable degree?

Note that it is well-known that X is reflexive if and only if π is separable (see [7, (4.4)]), and that the answer for the case n = 1 is known to be affirmative (see [2], [4], [5], [10]).

Related to this problem, although it has a slightly different flavour from the subject of the present paper, we include the following result here since it is proved by a quite similar way to the case of Proposition 1:

Proposition 2. Let Y ⊆ ℙ^{N} be a projective variety of dimension n ≥ 2, and let v_{d} be the Veronese embedding of ℙ^{N} of degree d ≥ 2. Then the Gauss map γ of v_{d}(Y) is generically finite and separable, and moreover it is finite and unramified if Y is smooth.

In their paper [8], Kleiman and Piene assert that in characteristic 2 the Gauss map of every variety of odd dimension is of inseparable degree at least 2 (see [8, pp. 109–110]). However, their assertion is obviously not correct.

The contents of this article are organized as follows: In §1 we give basic definitions and state a known fact due to Hefez and Thorup [3]. In §2 we show Propositions 1 and 2, with introducing some notation. In §3 we study the rank of symmetric matrices of certain form (see Lemma in §3), which plays the key role in our proof of Theorem. Finally in §4 we prove Theorem.

To simplify the notation and arguments, we work over an algebraically closed field K. But, all the results here are easily verified to be true over an arbitrary field by trivial modification of the setup and proofs given here: In that case, Y in Proposition 2 should be a closed subscheme of ℙ^{N} geometrically integral over K, and smooth over K in the latter assertion.

1. Preliminaries

For a projective variety X ⊆ ℙ^{N}, let CX^{∘} be the set of pairs (P, H) such that P is a smooth point of X and H is a hyperplane tangent to X at P. The conormal variety of X, denoted by CX, is by definition the closure of CX^{∘} in X × ℙ̌^{N}, and the dual variety of X, denoted by X^{∗}, is the image of CX in ℙ̌^{N} under the natural projection, where ℙ̌^{N} is the dual space of ℙ^{N}. On the other hand, the Gauss map of X, denoted by γ, is by definition the rational map X −−→ 𝔾(n, ℙ^{N}) that sends a smooth point P of X to the embedded tangent space to X at P, where n is the dimension of X and 𝔾(n, ℙ^{N}) is the Grassmann variety of n-dimensional linear spaces in ℙ^{N}.

A projective variety X ⊆ ℙ^{N} is said to be reflexive if we have CX ≃ C(X^{∗}) via ℙ^{N} × ℙ̌^{N} ≃ ℙ̌^{N} × ℙ^{N}, and ordinary if X is reflexive and X^{∗} has codimension 1.

Moreover, for a general (hence, smooth) point P of X and for a rational function h on ℙ^{N} defining a general hyperplane tangent to X at P by its zero locus, denote by H(X) the Hessian matrix of the function h_{X} on X at P, where h_{X} is the restriction of h to X. In general we have the following inequalities:

1 ≤ codim X^{∗} ≤ cork H(X) + 1.

A projective variety X ⊆ ℙ^{N} is said to be semi-ordinary if H(X) has corank 1. For full details on those notions, we refer to [2], [6], [7], [8].

Fact (Hefez-Thorup [3, Lemma 3]). Let X ⊆ ℙ_{ℤ}^{N} be a closed, integral subscheme, smooth over ℤ, let K be an algebraically closed field, and denote by X_{K} the fibre product X ×_{ℤ} K. Then we have:

(1) codim(X_{K})^{∗} = cork H(X_{ℂ}) + 1.

(2) X_{K} is reflexive if and only if cork H(X_{K}) = cork H(X_{ℂ}).

Remark. Hefez and Thorup [3] proved a corresponding statement for an arbitrary field K with more general setup.

2. The Gauss map

Proof of Proposition 1. Let (x_{i1}, x_{i2}, …, x_{in_{i}}) = (1 : x_{i1} : x_{i2} : ⋯: x_{in_{i}}) be affine coordinates of ℙ^{n_{i}} for each i. For P = (x_{ia})_{1≤i≤r,1≤a≤n_{i}} ∈ ∏_{i=1}^{r} ℙ^{n_{i}}, the Segre embedding σ is given by

σ(P) = (1 : x_{11} : x_{12} : ⋯: x_{1n_{1}} : x_{21} : ⋯: x_{rn_{r}} : x_{11}x_{21} : ⋯: x_{1n_{1}} ⋯ x_{rn_{r}}) ∈ ℙ^{N},

and the embedded tangent space at σ(P) is spanned by n + 1 row vectors as follows:

σ(P), \frac{∂σ}{∂x_{11}} (P), …, \frac{∂σ}{∂x_{1n_{1}}} (P), \frac{∂σ}{∂x_{21}} (P), …, \frac{∂σ}{∂x_{rn_{r}}} (P),

where N +1 = ∏_{i=1}^{r}(n_{i} +1) and n = ∑_{i=1}^{r} n_{i}. Consider an (n+1)×(N +1)-matrix formed by those row vectors . If we denote by Δ(j_{0}, …, j_{n}) its minor of degree n + 1 consisting of columns j_{0}, …, j_{n}, then the Gauss map γ is given by

γ(P) = (Δ(j_{0}, …, j_{n}))_{1≤j_{0}<···<j_{n}≤N+1} ∈ ℙ^{M}

via the Plücker embedding, where M+1 = \binom{N+1}{n+1}. Then we see that Δ(1, …, n, n+1) = 1, and moreover that for any i with 1 ≤ i ≤ r and for any a with 1 ≤ a ≤ n_{i}, there exist integers l and m such that

Δ(1, 2, …, l − 1, m, l + 1, …, n + 1) = x_{ia},

where 1 ≤ l ≤ n + 1 < m ≤ N + 1: Indeed, for such i and a, there exists a column vector in the (n + 1) × (N + 1)-matrix whose coordinates are given by

(x_{ia}x_{jb}, 0, …, 0, x_{jb}, 0, …, 0, x_{ia}, 0, …)

with i < j or

(x_{jb}x_{ia}, 0, …, 0, x_{ia}, 0, …, 0, x_{jb}, 0, …)

with j < i. Thus, we see that γ is an embedding. □

Proof of Proposition 2. Let P_{0} be a smooth point of Y, let {x_{1}, …, x_{n}} be local parameters of Y at P_{0}, and denote by ι the embedding Y ↪ ℙ^{N}: one may assume that ι around P_{0} is given by

ι(x_{1}, …, x_{n}) = (1 : x_{1} : ⋯: x_{n} : ⋯) ∈ ℙ^{N},

so that v_{d} ∘ ι is given by

v_{d} ∘ ι(x_{1}, …, x_{n}) = (1 : x_{1} : ⋯: x_{n} : q_{1} : ⋯: q_{M} : ⋯) ∈ ℙ^{N_{d}},

where q_{1}, …, q_{M} are the monomials in the x_{i} of degree 2, M = \binom{n+1}{2} and N_{d}+1 = \binom{N+d}{d}. Then, we see that the Gauss map of v_{d}(Y) is locally an embedding around P_{0}: Indeed, a similar argument to the last part of the proof of Proposition 1 works with choosing reduced monomials from the q_{i}, where we need the assumption n ≥ 2. □

3. Symmetric matrices

Let n_{1}, …, n_{r} be positive integers, and set n := ∑_{i=1}^{r} n_{i}. Let M_{kl} be an n_{k} × n_{l}-matrix with k < l such that each component is an indeterminate, set M_{lk} :=^{t} M_{kl}, and consider an n × n-matrix as follows:

M(n_{1}, …, n_{r}) := [ \begin{matrix}0 & M_{12} & M_{13} & ⋯ & M_{1r} \\ M_{21} & 0 & M_{23} & ⋯ & M_{2r} \\ M_{31} & M_{32} & 0 & ⋯ & M_{3r} \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ M_{r1} & M_{r2} & M_{r3} & ⋯ & 0 \end{matrix} ] .

Denote by z_{ij} the (i, j)-th component of M(n_{1}, …, n_{r}) with i < j. Then M(n_{1}, …, n_{r}) is a symmetric matrix over a polynomial ring ℤ[{z_{ij}|1 ≤ i < j ≤ n}].

Lemma. For given positive integers n_{1}, …, n_{r} with n := ∑_{i=1}^{r} n_{i}, and for an arbitrary field K of characteristic p ≥ 0, we have:

(1) If 2n_{i} > n, then M(n_{1}, …, n_{r}) has corank 2n_{i} − n over K.

(2) Assume 2n_{i} ≤ n for all i. If p = 2 and n is odd, then M(n_{1}, …, n_{r}) has corank 1 over K; Otherwise, M(n_{1}, …, n_{r}) has corank 0 over K.

Proof. (1) Renumbering the indexes, one may assume that 1 ≤ n_{1} ≤ n_{2} ≤ ⋯ ≤ n_{r}. Consider a partition of M(n_{1}, …, n_{r}) as follows:

M(n_{1}, …, n_{r}) = [ \begin{matrix}M′_{11} & M′_{12} & M′_{13} \\ M′_{21} & 0 & 0 \\ M′_{31} & 0 & 0 \end{matrix} ] ,

where M′_{11} and M′_{12} are (n−n_{r})×(n−n_{r})-matrices, M′_{13} is an (n−n_{r})×(2n_{r} −n)-matrix and M′_{lk} =^{t} M′_{kl}. Then the first principal minor of degree 2(n − n_{r}) is equal to

(−1)^{n−n_{r}} det M′_{12} det M′_{21} = (−1)^{n−n_{r}}(det M′_{12})^{2},

whose coefficient in

z_{1,n−n_{r}+1}^{2}z_{2,n−n_{r}+2}^{2} ⋯ z_{n−n_{r},2n−2n_{r}}^{2}

is equal to ±1. On the other hand, any minor of degree 2(n − n_{r}) + 1 is obviously zero. Therefore, M(n_{1}, …, n_{r}) has rank 2(n − n_{r}) over ℚ as well as over ℤ/pℤ for all prime p, hence over an arbitrary field K.

(2) It suffices to show:

(a) for odd n = 2m + 1, all the coefficient of det M(n_{1}, …, n_{r}) are divisible by 2, and there exist integers i(1), …, i(m), j(1), …, j(m), k with i(t) < j(t) for all t and j(m) < k such that

{i(1), …, i(m), j(1), …, j(m), k} = {1, …, n}

and the coefficient of det M(n_{1}, …, n_{r}) in

z_{i(1)j(1)}^{2} ⋯ z_{i(m−1)j(m−1)}^{2}z_{i(m)j(m)}z_{i(m)k}z_{j(m)k}

is equal to ±2.

(b) for even n = 2m, there exist integers i(1), …, i(m), j(1), …, j(m) with i(t) < j(t) for all t such that

{i(1), …, i(m), j(1), …, j(m)} = {1, …, n}

and the coefficient of det M(n_{1}, …, n_{r}) in

z_{i(1)j(1)}^{2} ⋯ z_{i(m)j(m)}^{2}

is equal to ±1; Indeed, if p ≠ 2 or n is even, then it clearly follows from (a) and (b) that M(n_{1}, …, n_{r}) has rank n over an arbitrary field K. If p = 2 and n is odd, then it follows from (a) that M(n_{1}, …, n_{r}) has rank at most n − 1. Moreover we see from (b) that the rank is actually equal to n − 1, as follows: Renumbering the indexes, we may assume that n_{1} ≥ n_{2} ≥ ⋯ ≥ n_{r} ≥ 1. Consider an (n − 1) × (n − 1)-matrix M′ by taking off the 1-st row and column from M(n_{1}, …, n_{r}), which has the same form as M(n_{1} − 1, n_{2}, …, n_{r}). Then it follows from (b) that M′ has rank n − 1: Indeed, the sizes n_{1} − 1, n_{2}, …, n_{r} of blocks in M′ satisfy the assumption of (2), where we omit n_{1} − 1 if it is zero. To see this, it suffices to verify the condition for n_{2}: this is trivial for n_{1} − 1 by the assumption of (2). Suppose the contrary, that is, suppose

n_{2} > (n_{1} − 1) + n_{3} + ⋯ + n_{r}.

Then we have n_{2} ≥ n_{1}+n_{3} ≥ n_{1}, so that n_{1} = n_{2} and n_{3} = 0 with r = 2, which contradicts to our assumption n being odd.

Now, let us show the statements (a) and (b). To show the former assertion of (a), denote by 𝔖_{n} be the permutation group of the set {1, 2, …, n}, and set

F := {g ∈ 𝔖_{n}|∃i, g(i) = i},
G_{+} := {g ∈ 𝔖_{n} \ F |#{i|g(i) > i} > n/2},
G_{−} := {g ∈ 𝔖_{n} \ F |#{i|g(i) > i} < n/2}.

Since n is odd, we have 𝔖_{n} = F ⊔ G_{+} ⊔ G_{−} and

det M(n_{1}, …, n_{r}) = ∑_{g∈𝔖_{n}} sign(g)z_{1g(1)}z_{2g(2)} ⋯ z_{ng(n)}

= ∑_{g∈F}(⋯) + ∑_{g∈G_{+}}(⋯) + ∑_{g∈G_{−}}(⋯).

Now, the sum over F is zero since the diagonal components of M(n_{1}, …, n_{r}) are zero. On the other hand, the sum over G_{−} is equal to the sum over G_{+} since M(n_{1}, …, n_{r}) is symmetric: Indeed, since a map G_{+} → G_{−}; g ↦ g^{−1} is bijective and sign(g) = sign(g^{−1}), we have

∑_{g∈G_{−}} sign(g)z_{1g(1)}z_{2g(2)} ⋯ z_{ng(n)} = ∑_{g∈G_{+}} sign(g^{−1})z_{1g^{−1}(1)}z_{2g^{−1}(2)} ⋯ z_{ng^{−1}(n)}

= ∑_{g∈G_{+}} sign(g)z_{g(1)1}z_{g(2)2} ⋯ z_{g(n)n},

which is equal to the that sum over G_{+} since z_{g(i)i} = z_{ig(i)}. This proves the former assertion of (a).

For the remaining part, we use induction on n, where we allow the integers n_{i} being zero. Assume n ≥ 4: Note that the determinant is equal to −z_{12}^{2} for n = 2, and equal to 2z_{12}z_{13}z_{23} for n = 3. Assume that n_{1} ≥ n_{2} ≥ ⋯ ≥ n_{r} ≥ 1, as before, and consider an (n − 2) × (n − 2)-matrix M′′ by taking off the 1-st and (n_{1} + 1)-th rows and columns from M(n_{1}, …, n_{r}), which has the same form as M(n_{1} − 1, n_{2} − 1, n_{3}, …, n_{r}). Then, for M′′ we may assume the hypothesis of induction: Indeed, the sizes n_{1} − 1, n_{2} − 1, n_{3}, …, n_{r} of blocks of M′′ satisfy the assumption of (2). To see this, it suffices to verify the condition for n_{3} with r ≥ 3, as before. Suppose the contrary, that is, suppose

n_{3} > (n_{1} − 1) + (n_{2} − 1) + n_{4} + ⋯ + n_{r}.

Then we have n_{3} ≥ n_{1} + n_{2} − 1 ≥ 2n_{3} − 1, so that n_{3} = 1, which implies n_{1} = n_{2} = 1 with r = 3 and n = 3. This contradicts to our assumption n ≥ 4.

Now, it follows from the hypothesis of the induction that there exist integers i(2), …, i(m), j(2), …, j(m), k satisfying

{i(2), …, i(m), j(2), …, j(m), k} = {1, …, n} \ {1, n_{1} + 1}

for n = 2m + 1 (respectively, there exist integers i(2), …, i(m), j(2), …, j(m) satisfying

{i(2), …, i(m), j(2), …, j(m)} = {1, …, n} \ {1, n_{1} + 1}

for n = 2m) such that the above property on the coefficient for M′′ is satisfied. Then for n = 2m + 1 the coefficient of det M(n_{1}, …, n_{r}) in

z_{1n_{1}+1}^{2}z_{i(2)j(2)}^{2} ⋯ z_{i(m−1)j(m−1)}^{2}z_{i(m)j(m)}z_{i(m)k}z_{j(m)k}

is equal to ±2, and for n = 2m the coefficient of det M(n_{1}, …, n_{r}) in

z_{1n_{1}+1}^{2}z_{i(2)j(2)}^{2} · · · · · · z_{i(m)j(m)}^{2}

is equal to ±1, as is required. □

4. The Hessian matrix

Let P_{0} be the point of the Segre variety X with all x_{ia} being zero, and let h be a rational function on ℙ^{N} corresponding to a general hyperplane tangent to X at P_{0}, with the same notation as in §2. Then we have

σ^{∗}h = ∑_{1≤i<j≤r,1≤a≤n_{i},1≤b≤n_{j}} h_{ij}^{ab}x_{ia}x_{jb} + (terms of higher degree in x_{kc}),

where the h_{ij}^{ab} are general elements of K. Therefore, the Hessian matrix at P_{0} is given as follows:

H(X) = [\frac{∂^{2}σ^{∗}h}{∂x_{ia}∂x_{jb}} (P_{0})] = [ \begin{matrix}0 & H_{12} & H_{13} & ⋯ & H_{1r} \\ H_{21} & 0 & H_{23} & ⋯ & H_{2r} \\ H_{31} & H_{32} & 0 & ⋯ & H_{3r} \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ H_{r1} & H_{r2} & H_{r3} & ⋯ & 0 \end{matrix} ] ,

where H_{ij} = [ h_{ij}^{ab}]_{1≤a≤n_{i},1≤b≤n_{j}} and H_{ji} =^{t} H_{ij} with 1 ≤ i < j ≤ r.

Proof of Theorem. (1) Using Fact (1), we see that the if-part follows from Lemma (2), and the only-if-part follows from Lemma (1).

(2) The assertion (a) follows from Fact (2) and Lemma (1), and (b) also follows from Lemma (1).

(3) Note that since codim X^{∗} = 1 by (1), X is reflexive if and only if X is ordinary. Then, the equivalence (a) ⇔ (c) follows from Fact (2) and Lemma (2), and (b) ⇔ (c) also follows from Lemma (2). □

Acknowledgements. The results here were announced in the Fourth Meeting of the Brazilian Group in Commutative Algebra and Algebraic Geometry, ALGA-2001, organized by Arnaldo Garcia, Eduardo Esteves and Amílcar Pacheco, held in Angra dos Reis, Rio de Janeiro, August 5–11, 2001. This work was done while the author stayed in IMPA (=Instituto Nacional de Matemática Pura e Aplicada). He wishes to thank the people at IMPA, especially Prof. Arnaldo Garcia and Prof. Eduardo Esteves for their warm hospitality. Muito obrigado!

References

1. I. M. Gelfand, M. M. Kapranov, A. V. Zelevinsky, Hyperdeterminants, Adv. Math. 96 (1992), 226–263.

2. A. Hefez, S. Kleiman, Notes on the duality of projective varieties, “Geometry Today”, Prog. Math., vol. 60, Birkhäuser, Boston, 1985, pp. 143–183.

3. A. Hefez, A. Thorup, Reflexivity of Grasmannians and Segre verieties, Comm. Alg. 15 (6) (1987), 1095–1108.

4. A. Hefez, J. Voloch, Frobenius non classical curves, Arch. Math. 54 (1990), 263–273.

5. H. Kaji, On the Gauss maps of space curves in characteristic p, Compos. Math. 70 (1989), 177–197.

6. H. Kaji, On the inseparable degrees of the Gauss map and the projection of the conormal variety to the dual of higher order for space curves, Math. Ann. 292 (1992), 529–532.

7. S. Kleiman, About the conormal scheme, Springer Lecture Notes in Math. 1092 (1984), 161–197.

8. S. Kleiman, R. Piene, On the inseparability of the Gauss map, Contemp. Math. 123 (1991), 107–129.

9. F. Knop, G. Menzel, Duale Varietäten von Fahnenvarietäten, Comment. Math. Helvetici 62 (1987), 38–61.

10. J. Voloch, On the geometry of space curves, Proceedings of the 10th School of Algebra, Vitoria, Brazil, 1989, pp. 121–122.

(September 2000–August 2001) Instituto Nacional de Matemática Pura e Aplicada, Estrada Dona Castorina, 110, 22460–320 Rio de Janeiro RJ, BRASIL

E-mail address: kaji@impa.br

Department of Mathematical Sciences, School of Science and Engineering, Waseda University, Tokyo 169–8555, JAPAN

  E-mail address: kaji@mse.waseda.ac.jp