Complete intersections in relation to a paper of B.J.Birch

A.G.Aleksandrov^{∗}, B.Z.Moroz

Abstract. We prove a new property of complete intersections in graded Noetherian rings, and relate it to the Jacobian criterion. We then apply these results to describe the loci of singularities in a family of affine varieties defined by B.J.Birch in his classic paper on forms of many variables. AMS Classification: Primary 13A02, 13C40, 11E76, 11D72; Secondary 13D25, 13C15, 14M10, 11G35 Key words and phrases: graded rings; complete intersection; regular sequences; Jacobian criterion; singularities; unmixedness theorem; Cohen-Macaulay rings and modules; homogeneous forms; rational points

1. An assertion of B.J.Birch

In a famous memoir, Professor B.J.Birch concludes as follows (see [2], p.245): Let f_{1}, …, f_{R} be rational forms of degree d in n variables x_{1}, …, x_{n}, where

n > R ≥ 1. Let V (μ) be the variety

V (μ): f_{1}(x) = μ_{1}, …, f_{R}(x) = μ_{R},  (1)

and let V^{∗} be the union of the loci of singularities of the V (μ) (the μ_{i} run through some algebraically closed field). Then if dim V (0) = n − R, and if V (0) has non-singular real points and non-singular points in every p-adic completion of the rationals, and if

n − dim V^{∗} > R(R + 1)(d − 1)2^{d−1},

then V (0) certainly has a rational point other than the origin.

As one of the most general statements asserting existence of rational points on a projective algebraic variety, this theorem has been widely cited. An assiduous reader will note however that, in the course of the proof, Professor Birch makes use of the following assertion, [2], p.250 :

∗Partially supported by the Russian Foundation of Fundamental Research (RFFR), Grant No. 99-01-01067.

V^{∗} consists of all points x for which

rank | \frac{∂f_{i}}{∂x_{j}} |_{i,j} < R.  (2)

Far from being obvious, this assertion does not seem to have been proved in the literature; some of the authors, [4], [7], [8], just define V^{∗} by the condition (2), while other authors do not feel the cited assertion needs any justification, see, for instance, [9], where Birch’s theorem has been generalized to treat projective varieties defined over an algebraic number field. In this note, we prove that this assertion is indeed correct.

To be precise, equations (1) define an affine scheme

V (μ) = Spec k[x]/𝔄_{μ} ,

where k is a fixed (algebraically closed field) and 𝔄_{μ} = (f_{1}(x) − μ_{1}, …, f_{R}(x) − μ_{R}) is the ideal in k[x] corresponding to these equations. In general, this scheme is neither reduced, nor irreducible. Condition (2) defines the singular locus of V (μ) if (i) dim V (μ) = n − R, and (ii) the Jacobian criterion may be applied. Making use of the tools of modern commutative algebra, [3], [5], [6], we show that each of these conditions follows from the assumption dim V (0) = n − R.

Let J_{R} be the ideal in k[x] generated by the minors of order R of the Jacobian matrix

( \frac{∂f_{j}}{∂x_{i}} )_{{1≤j≤R \atop 1≤i≤n}} ,

and let W = Spec k[x]/J_{R}. Theorem 1. Let f_{j}(x), 1 ≤ j ≤ R, be non-constant homogeneous polynomials of the same degree. If dim V (0) = n − R and k is a perfect field, then

V^{∗} = Spec k[x]/ℭ

with J_{R} ⊆ ℭ ⊆ \sqrt{J_{R}}.

This theorem follows from our main result, Theorem 2, and will be proved in §4. As an immediate consequence of the theorem, one obtains the following result.

Corollary 1.

(i) The set V^{∗} is homeomorphic to W, and in particular dim V^{∗} = dim W. (ii) The set of k-rational points V^{∗}(k) of V^{∗} coincides with the set of k-rational

points W (k) of W, in accordance with the assertion of B.J.Birch cited above.

In the next section, Theorem 2 is stated and proved; it describes a new property of complete intersections in graded Noetherian rings.

2. Complete intersections and regular sequences

To establish our terminology, let us remind that a finite subset I of a commutative ring A defines a complete intersection (see [3], §9, n^{o}6), if H_{i}(I, A) = 0 for i > 0, where H_{i}(I, A) denotes the i-th homology group of the Koszul complex associated to the set I. Further, a sequence {a_{1}, …, a_{ℓ}} of elements of A is said to be regular if a_{1} is not a zero divisor of the ring A and if for any 1 ≤ j ≤ ℓ − 1 the element a_{j+1} is not a zero divisor in the quotient A/(a_{1}, …, a_{j}). The property to be regular depends on the ordering of the elements of the sequence, while the property to define a complete intersection does not. In this section N stands for set of non-negative rational integers. Lemma 1. Let {a_{1}, …, a_{ℓ}} be a sequence of elements in A, and let (b_{ij}) ∈ GL_{ℓ} (A), 1 ≤ i, j ≤ ℓ. Set

y_{i} = ∑_{j=1}^{ℓ} b_{ij}a_{j}, 1 ≤ i ≤ ℓ.

Then the sequence {a_{1}, …, a_{ℓ}} defines a complete intersection if and only if the sequence {y_{1}, …, y_{ℓ}} does.

Proof. See [3], §9, n^{o}6, Prop.6, b).

Lemma 2. If {a_{1}, …, a_{ℓ}} is a regular sequence, then the set {a_{j} : 1 ≤ j ≤ ℓ} defines a complete intersection.

Proof. See [3], §9, n^{o}7, Th.1.

The converse of Lemma 2 is true in the following situation.

Lemma 3. Let A be a Noetherian N-graded ring, A ≠ 0. If the set {a_{j}: 1 ≤ j ≤ ℓ} defines a complete intersection and if each of the elements a_{j} of this set is homogeneous of positive degree, then the sequence {a_{1}, …, a_{ℓ}} is regular.

Proof. See [6], Theorem 16.5 (β).

As a consequence of Lemmata 1-3, we obtain the following result.

Theorem 2. Let A be a Noetherian N-graded ring, and let a_{j}, 1 ≤ j ≤ ℓ, be homogeneous elements of degree d , d ≥ 1. If the set {a_{1}, …, a_{ℓ}} defines a complete intersection, then for any μ ∈ (A^{∗} ∪ {0})^{ℓ} the set {a_{j} + μ_{j}: 1 ≤ j ≤ ℓ} defines a complete intersection; here μ = (μ_{1}, …, μ_{ℓ}), and A^{∗} stands for the set of units of A.

Proof. In view of Lemma 1, the set {a_{1} + b_{1}a_{ℓ}, …, a_{ℓ−1} + b_{ℓ−1}a_{ℓ}, a_{ℓ}} defines a complete intersection for any b_{i} ∈ (A^{∗} ∪ {0}), i = 1, …, ℓ − 1. By Lemma 3, the sequence {a_{1} + b_{1}a_{ℓ}, …, a_{ℓ−1} + b_{ℓ−1}a_{ℓ}, a_{ℓ}} and, consequently, its subsequence {a_{1} + b_{1}a_{ℓ}, …, a_{ℓ−1} + b_{ℓ−1}a_{ℓ}} are regular. Suppose, without loss of generality, μ_{ℓ} ≠ 0. Then a_{ℓ} +μ_{ℓ} is not a zero divisor in the ring A/(a_{1} +b_{1}a_{ℓ}, …, a_{ℓ−1} +b_{ℓ−1}a_{ℓ}); therefore the sequence {h_{1}, …, h_{ℓ}}, with h_{i} = a_{i} + b_{i}a_{ℓ}, 1 ≤ i ≤ ℓ − 1, h_{ℓ} = a_{ℓ} + μ_{ℓ} is regular. By Lemma 2, it defines a complete intersection; therefore (Lemma 1 !) the set {h_{1} − b_{1}h_{ℓ}, …, h_{ℓ−1} − b_{ℓ−1}h_{ℓ}, h_{ℓ}} defines a complete intersection. It now suffices to choose b_{i} = −μ_{i}/μ_{ℓ}, 1 ≤ i ≤ ℓ − 1. This concludes the proof of the theorem.

Remark 1. One may note that to define a complete intersection is an open property in the sense of A.Grothendieck (see, for instance, [1]). Thus, a priori, this property holds for those μ outside a closed subset. In the theorem, we have given some conditions to ensure it is valid for every value of the parameters μ.

3. Jacobian criterion and singular locus

Let us remind that the depth of an ideal 𝔅, to be denoted by dp 𝔅, is equal to the maximal possible length of a regular sequence contained in 𝔅. The height of an ideal 𝔅, to be denoted by ht 𝔅, is defined to be the minimum of the heights of the prime ideals containing 𝔅. In a Cohen-Macaulay ring, dp 𝔅 = ht 𝔅. Finally, an ideal 𝔅 is said to have a pure codimension r if ht ℘ = r for any minimal prime ℘ containing 𝔅. Note that in [5] height is called codimension and is denoted by “codim”.

In what follows, we write H_{i}(𝔅, A) for H_{i}(I, A), where 𝔅 is the ideal in A generated by the set I. We shall make use of the following theorems of commutative algebra. Lemma 4. Let 𝔅 be an ideal in an integral domain A. Then

dim A/𝔅 + ht 𝔅 = dim A.

Proof. See [5], Corollary 13.4, p.286.

Lemma 5. Let 𝔅 = (a_{1}, …, a_{ℓ}) be an ideal in a Noetherian ring A, and suppose 𝔅 ≠ A. Then the sequence {a_{1}, …, a_{ℓ}} is regular if and only if dp 𝔅 = l.

Proof. See [6], Corollary on p.131.

Lemma 6. Let 𝔅 = (a_{1}, …, a_{ℓ}) be an ideal in a Noetherian ring A, suppose 𝔅 ≠ A, and let

q = sup{i : H_{i}(𝔅, A) ≠ 0}.

Then any maximal regular sequence contained in 𝔅 has length n − q.

Proof. See [6], Theorem 16.8, p. 131.

To deduce Theorem 1 from Theorem 2 we shall make use of the Jacobian criterion in the following form (see [5], Corollary 16.20).

Lemma 7. Let A = k[x_{1}, …, x_{n}]/I be an affine ring over a perfect field k and suppose that I is of pure codimension r; let I = (g_{1}, …, g_{ℓ}). If J is the ideal of A generated by the r × r minors of the Jacobian matrix (∂g_{j}/∂x_{i}), then J defines the singular locus of A in the sense that a prime ℘ of A contains J if and only if A_{℘} is not a regular local ring.

Proposition 1. In the notation of §1, if dim V (0) = n − R then the ideal 𝔄_{μ} has a pure codimension R for any μ in k^{R}.

Proof. By Lemma 4, condition dim V (0) = n−R shows that ht 𝔄_{0} = R. Since k[x] is a Cohen-Macaulay ring, it follows that dp 𝔄_{0} = ht 𝔄_{0} = R. Further, 𝔄_{0} ≠ k[x], therefore by Lemma 5 the sequence {f_{1}, …, f_{R}} is regular, and Lemma 3 implies that the set {f_{j} : 1 ≤ j ≤ R} defines a complete intersection. Thus we can apply Theorem 2 to conclude that the set {f_{j} − μ_{j} : 1 ≤ j ≤ R} defines a complete intersection for any μ. Moreover,

H_{0}(𝔄_{μ}, k[x]) ≅ k[x]/𝔄_{μ} ≠ 0,

therefore it follows from Lemma 6 that dp 𝔄_{μ} = R; consequently, ht 𝔄_{μ} = R. An application of the Macaulay unmixedness theorem [5], Corollary 18.14, concludes the proof of the proposition.

4. The proof of the main theorem

Suppose now k be a perfect field. It follows from Proposition 1 and Lemma 7 that

Sing V (μ) = Spec k[x]/(𝔄_{μ}, J_{R}),

where Sing V (μ) denotes the singular locus of V (μ). By definition,

V^{∗} = ⋃_{μ∈k^{R}} V (μ).

Let ℭ = ∩_{μ∈k^{R}}(𝔄_{μ}, J_{R}), then

V^{∗} = Spec k[x]/ℭ.

Clearly, J_{R} ⊆ ℭ. Let h ∈ ℭ, u ∈ k^{n}, and suppose g(u) = 0 for every g in J_{R}. For any μ in k^{R}, one may write

h = ∑_{j=1}^{R} a_{j}^{(μ)}(f_{j} − μ_{j}) + b_{μ},

where a_{j}^{(μ)} ∈ k[x], b_{μ} ∈ J_{R}. Then

h(u) = ∑_{j=1}^{R} a_{j}^{(μ)}(u)(f_{j}(u) − μ_{j}),

therefore h(u) = 0 since we may take μ_{j} = f_{j}(u), 1 ≤ j ≤ R, and Hilbert’s Nullstellensatz shows that h ∈ \sqrt{J_{R}}. Thus

J_{R} ⊆ ℭ ⊆ \sqrt{J_{R}},

as asserted in Theorem 1.

Acknowledgment. It is a pleasure to thank the Max-Planck-Institut für Mathematik for its financial assistance and hospitality during the preparation of this paper.

References

[1] J. Bingener, H. Flenner, On the fibers of analytic mappings. In: Complex analysis and geometry, Univ. Ser. Math., Plenum, New York, 1993, 45-101.

[2] B. J. Birch, Forms in many variables, Proc. Roy. Soc., Ser. A 265 (1961/1962), 245-263.

[3] N. Bourbaki, Algèbre homologique, Masson, Paris, 1980.

[4] R. Danset, Méthode du cercle adélique et principe de Hasse fin pour certains systèmes de formes. Enseign. Math. (2) 31 (1985), no. 1-2, 1-66.

[5] D. Eisenbud, Commutative algebra. With a view toward algebraic geometry. Graduate Texts in Mathematics 150. Springer-Verlag, New York, 1995.

[6] H. Matsumura, Commutative ring theory. Cambridge Studies in Advanced Mathematics 8. Cambridge University Press, Cambridge-New York, 1986.

[7] B. Z. Moroz, On solubility of Diophantine equations (a survey of some results from the analytic theory of numbers). Max-Planck-Institut für Mathematik, Preprint series 1999, no. 118, submitted for publication.

[8] W. M. Schmidt, The density of integer points on homogeneous varieties. Acta Math. 154 (1985), no. 3-4, 243-296.

[9] C. M. Skinner, Forms over number fields and weak approximation. Compositio Math. 106 (1997), no. 1, 11-29.

A.G.Aleksandrov  B.Z.Moroz
Institute of Control Sciences  Max-Planck-Institut für Mathematik
Russian Academy of Sciences  Vivatsgasse 7
Profsojunaja ul. 65  D-53111, Bonn,
GSP-7, Moscow, 117806  Germany
Russian Federation  E-mail: moroz@mpim-bonn.mpg.de

E-mail: alexandr@ipu.rssi.ru

Current address: Max-Planck-Institut für Mathematik Vivatsgasse 7 D-53111, Bonn Germany E-mail: alexandr@mpim-bonn.mpg.de