The linear subspace section of variety by specializations

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In this article, we will prove the preservation of some properties of the generic linear subspace sections of nondegenerate varieties by specializations and ground forms of components of variety are conjugated.

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JOURNAL OF SCIENCE OF HNUE Mathematical and Physical Sci., 2012, Vol. 57, No. 7, pp. 12-19 This paper is available online at http://stdb.hnue.edu.vn THE LINEAR SUBSPACE SECTION OF VARIETY BY SPECIALIZATIONS Dam Van Nhi School for Gifted Students, Hanoi National University of Education Abstract. In this paper, we prove the preservation of some properties of the generic linear subspace sections of nondegenerate varieties by specializations. Keywords: Specialization, variety, absolutely irreducible. 1. Introduction To study smooth curves one intersects the curve with a general hyperplane and studies the resulting finite set of points. The Harris’s lemma [2] about a set of points in the uniform position has attracted much attention in algebraic geometry. This is a set of points of a projective space such that any two subsets, each with the same cardinality, have the same Hilbert function. We bring up the question of whether other uniformed position properties remain unchanged when a curve is intersected by any hyperplane. To answer this question, we use the notation ground form which was given by E. Noether [8], B.L. van der Waerden [11] and specializations of modules and of graded modules which was given by D. V. Nhi and N. V. Trung [6, 7]. In this article, we will prove the preservation of some properties of the generic linear subspace sections of nondegenerate varieties by specializations and ground forms of components of variety are conjugated. Throughout this paper, Ω will be the universal field, which is algebraic and has an infinite degree of transcendence over an infinite ground field K. 2. Some results about specializations of modules Let K be an infinite ground field and K be the algebraic closure of K. Denote by u = (u1, u2 , . . . , us ), a system of s new indeterminates ui , which are algebraically independent of K, K[u, x] and the polynomial ring K[u1 , . . . , us , x1 , . . . , xn ]. We shall Received September 28, 2012. Accepted October 5, 2012. 2000 Mathematics Subject Classification: Primary 13A30, Secondary 13D45. Contact Dam Van Nhi, e-mail address: damvannhi@yahoo.com 12
The linear subspace section of variety by specializations P m1 P mn regard the ui as parameters over K. Let f (u, x) = ... ai1 ...in ud11 . . . uds s xi11 . . . xinn i1 =0 in =0 be any polynomial in K[u, x]. Let α = (α1 , . . . , αs ) be any element of Ωs . The polynomial m1 X mn X f (α, x) = ... ai1 ...in α1d1 . . . αsds xi11 . . . xinn i1 =0 in =0 is said to be a specialization of f with respect to the substitution u → α. We denote the polynomial rings in n variables x1 , . . . , xn over K(u) and K(α) by R = K(u)[x] and by Rα = K(α)[x], respectively. Each element a(u, x) of R can be p(u, x) written in the form a(u, x) = with p(u, x) ∈ K[u, x]andq(u) ∈ K[u] \ {0}. For q(u) any α with q(α) 6= 0 we define p(α, x) a(α, x) = . q(α) We shall say that a property holds for almost all α if it holds for all points of a Zariski-open non-empty subset of Ωm . For convenience we will generally omit the phrase "for almost all α" in the proofs of the results presented in this paper. Then we have the following lemmas: Lemma 2.1. [9, lemma 7] Let f (u, x) be a polynomial in K[u, x]. If f (u, x) is irreducible in K(u)[x] then f (α, x) is also irreducible for almost all α. Lemma 2.2. [9, lemma 8] If f (u, x) ∈ K(u)[x] is a power of an irreducible polynomial in K(u)[x] then abvef (α, x) is the same power of an irreducible polynomial in K(α)[x] for almost all α. Lemma 2.3. [1, 2.Satz] If f (u, x) is an absolute irreducible polynomial in K[u, x] then f (α, x) is also an absolute irreducible polynomial for almost all α. Following [10] we define the specialization of an ideal I of a polynomial ring R = K(u)[x] with respect to the substitution u −→ α as the ideal Iα of Rα = K(α)[x] generated by elements of the set {f (α, x)| f (u, x) ∈ I ∩ k[u, x]}. This is obviously an ideal of the polynomial ring Rα . For almost all substitutions u −→ α, the ideal Iα inherits most of the basic properties of I. The specialization of ideals can be generalized to modules. Let F be a free R-module of finite rank. The specialization Fα of F is a free Rα -module of the same rank. Let φ : F −→ G be a homomorphism of free R-modules. We can represent φ by a matrix A = (aij (u, x)) with respect to fixed bases of F and G. Let Aα = (aij (α, x)). Then Aα is well-defined for almost all α. The specialization φα : Fα −→ Gα of φ is given by the matrix Aα provided that Aα is well-defined. We note that the definition of φα depends on the chosen bases of Fα and Gα . 13
Dam Van Nhi φ Definition 2.1. [6] Let L be an R-module. Let F1 −→ F0 −→ L −→ 0 be a finite free presentation of L. Let φα : (F1 )α −→ (F0 )α be a specialization of φ. We call Lα := Coker φα a specialization of L (with respect to φ). If we choose a different finite free presentation F1′ −→ F0′ −→ L −→ 0 we may get a different specialization L′α of L, but Lα and L′α are canonically isomorphic. Hence Lα is uniquely determined up to isomorphisms and we have the result. Lemma 2.4. [6, theorem 2.4] Let 0 −→ L −→ M −→ N −→ 0 be an exact sequence of finitely generated R-modules. Then the sequence 0 −→ Lα −→ Mα −→ Nα −→ 0 is exact for almost all α. Let L be a graded R-module of dimension d. Let hL (t) denote the Hilbert polynomial of L. Then we have: Lemma 2.5. [7, Corollary 2.3] Let L be a graded R-module. Then, for almost all α, we have hLα (t) = hL (t), 3. Preservation of some properties of hyperplane sections of varieties by specializations In this section, we are interested in the hyperplane sections of varieties. Before reproving some results about hyperplane sections of varieties we recall the notation of a ground form which is introduced in order to study the properties of points on a variety. The concept of ground forms was formulated by E. Noether [8]. More accounts will be found in W. Krull [4] or B. L. van der Waerden [11]. Denote by (u) = (uij ) with i = 0, 1, . . . , n, and j = 0, 1, . . . , n, a system of (n + 1)2 new indeterminates uij , are algebraically independent over K. We enlarge K by adjoining (u). Consider the polynomial rings K(u)[x] = K(u)[x0 , x1 , . . . , xn ] and K(u)[y] = K(u)[y0 , y1, . . . , yn ]. The general linear transformation establishes an isomorphism between K(u)[x] and K(u)[y] when in every polynomial of K(u)[y] the substitution n X yi = uij xj , i = 0, 1, . . . , n, j=0 is carried out and the inverse transformation n X xi = vij yj , i = 0, 1, . . . , n, j=0 has its coefficients vij ∈ K(u). We get K(u)[x] = K(u)[y]. Every ideal I of K[x] generates an ideal IK(u)[x], which is transformed by the above isomorphism into the 14
The linear subspace section of variety by specializations ideal n X n X n X ∗ I = ({f ( v0j yj , v1j yj , . . . , vnj yj )|f (x) , x1 , . . . , xn ) ∈ I}). j=0 j=0 j=0 We consider an unmixed d-dimenisional homogeneous ideal P ⊂ K[x]. Then, P is transformed into the ideal P ∗ and the following ideal P ∗ ∩ K(u)[y0 , y1, . . . , yd ] = (F (u, y0, y1 , . . . , yd )) is a principal ideal of K(u)[y0 , . . . , yd]. We may suppose that F is normalized so as to be a polynomial in uij and primitive in them. By a linear projective transformation, we can choose F so that it is regular in y0 . The form F (u, y0, . . . , yd ) is called a ground form of P. It was well known that the ground form F of a prime ideal P is an irreducible form, but P is primary if and only if its ground form is a power of an irreducible form. We emphasize that if P1 and P2 are distinct d-dimensional prime ideals, then the ground form of P1 is not a constant multiple of the ground form of P2 , and the ground form of a d-dimensional ideal is a product of ground forms of d-dimensional primary componentes, see [4], Satz 3 and Satz 4. Given any homogeneous ideal I of the standard grading polynomial ring K[x] = L K[x0 , . . . , xn ] with deg xi = 1. We now set R = K[x]/I = t≥0 Rt . The Hilbert function of I, which is denoted by h(−; I), is defined as follows h(t; I) = dimK Rt for all t ≥ 0. We make a number of simple observations which are needed afterwards. It is easy to check the following result: Lemma 3.1. The Hilbert function is unchanged by projective inverse transformation. If K ∗ is an extension field of K, then h(t; I) = h(t; IK ∗ [x]) for all t ≥ 0. Now we want to study the hyperplane section of nondegenerate varieties. Definition 3.1. A variety V of Pn is nondegenerate if it does not lie in any hyperplane. Definition 3.2. A variety V of Pn defined over K is said to be absolutely irreducible if it is still irreducible when it is considered as a variety over any extension of K. L Put a = I(V ) = j≥1 aj . Note that V is nondegenerate if and only if a1 = 0 or h(1; a) = n + 1. Consider the intersection Wα = V ∩ Hα of an irreducible nondegenerate variety V with a hyperplane Hα : ℓα = α0 x0 + · · · + αn xn = 0, αi ∈ K ∗ . 15
Dam Van Nhi Proposition 3.1. Let V be an irreducible nondegenerate variety of Pn with d = dim V > 1. Then Wα = V ∩ Hα is again an irreducible nondegenerate variety of Pn−1 with dim Wα = dim V − 1 for almost all α. Proof. Denote Hv : ℓv = v0 x0 + · · · + vn xn = 0 the general hyperplane. Since the irreducibility of a variety is preserved by finite pure transcendental extension of ground field, therefore V is still an irreducible variety of dimension d over K(v). We have I(V ∩ Hv ) = (a, ℓv ) and V ∩ Hv is an irreducible variety of dimension d − 1 by Bertini’s theorem. Using a general transformation, the ground form of (a, ℓv ) can be assumed as a form F (u, v, y0, . . . , yd−1 ). By [9, theorem 6], F (u, α, y0, . . . , yd−1 ) is the ground form of (a, ℓα ) or of V ∩Hα . Since V ∩Hv is an irreducible variety, therefore F (u, v, x0, . . . , xd−1 ) is a power of an irreducible form. By lemma 2.2, F (u, α, x0, . . . , xd−1 ) is also a power of an irreducible form. Hence Wα is again an irreducible variety of dimension d − 1 for almost all α. Since V is nondegenerate, there is h(1; a) = n + 1. Because aK(v)[x] : ℓv = aK(v)[x], therefore aK(α)[x] : ℓα = aK(α)[x] for almost all α. Hence h(1; (aK(v)[x], ℓv )) = h(1; aK(v)[x]) − h(0; aK(v)[x]) = n + 1 − 1 = n. By lemma 2.5, h(1; (aK(α)[x], ℓα )) = h(1; aK(α)[x]) − h(0; aK(α)[x]) and we obtain h(1; I(Wα )) = n. Hence Wα = V ∩Hα is again an irreducible nondegenerate variety. Assume that V ⊂ Pn is an irreducible variety of dimension d over K. Denote I(V ) in K[x] by p. Then pK(u)[x] ∩ K(u)[y] = (F (u; y0, . . . , yd)) is a principal ideal. Note that if we consider V as an algebraic set over K then V may be a reducible set. Then we have the following result. Lemma 3.2. Let V be an irreducible variety of dimension d > 1 over K and Hα : S s α0 x0 + α1 x1 + · · · + αn xn = 0, αi ∈ K. Suppose that V = Vi is a decomposition of V i=1 into the irreducible varieties Vi over K. Then, every Vi ∩ Hα is an absolutely irreducible component of V ∩ Hα over K for almost all α. Proof. Denote Hv : ℓv = v0 x0 + v1 x1 + · · · + vn xn = 0 a general hyperplane, where Ss the vi are indeterminates. Assume that V = Vi is a decomposition of V into the i=1 irreducible varieties Vi over K. Since Vi is an irreducible variety over K, therefore Vi Ss is an absolutely irreducible variety of dimension d > 1. Hence V ∩ Hv = Vi ∩ Hv , i=1 where every Vi ∩ Hv is an absolutely irreducible variety by [3, X.13 theorem I]. Since Vi ∩ Hv is an absolutely irreducible variety, therefore the ground form Fi (u, v, y) of Pv = (I(Vi ), ℓv )K(v)[x] is a power an absolutely irreducible form f (u, v, x), an example 16
The linear subspace section of variety by specializations being Fi (u, v, y) = f (u, v, y)s. Since the gound form of Pα is Fi (u, α, y) = f (u, α, y)s and f (u, α, y) is also an absolutely irreducible form by lemma 2.3, Vi ∩ Hα is an irreducible variety over K. Hence, every Vi ∩ Hα is an absolutely irreducible variety for almost all α. Now we assume that V is a nondegenerate irreducible variety of dimension d over u K. Denote I(V ) by p. Suppose that the generic linear subspace Sn−d of dimension n − d is given by the generic linear equations Hiu : ℓiu = ui0 x0 + ui1 x1 + · · · + uin xn = 0, i = 1, 2, . . . , d. Put ui for the set of all uij with j = 0, 1, . . . , n and i = 0, 1, . . . , d. It was u well known that a generic linear subspace Sn−d meets V in a finite set of points T, and each of them is a generic point of V over K(u1 , . . . , ud ). The Harris’s lemma [2] shows that T has the Uniform Position Property. That is, any two subsets of T with the same cardinality have the same Hilbert function. The property about a set in the uniform position has attracted much attention in algebraic geometry. Consider a linear subspace α Sn−d of dimension n − d that is given by the linear equations Hiα : ℓiα = αi0 x0 + αi1 x1 + · · · + αin xn = 0, ∀αij ∈ K, i = 1, 2, . . . , d. The problem of concern in the following section is the existence of isomorphism α between round forms of components of the intersection V ∩ Sn−d . n Assume that V ⊂ P is an irreducible variety of dimension d over K. Let (ξ) = (ξ0 , ξ1 , . . . , ξn ) be a generic point of V such that K(ξ) is separable over K. Without loss of generality, we may suppose that this is normalized so that ξ0 = 1. The uij are algebraically P n independent over K(ξ1 , . . . , ξn ). We put λi = − uij ξj with i = 0, 1, . . . , d. Then j=0 λ1 , . . . , λd are algebraically independent over K(u), but λ0 , λ1 , . . . , λd are algebraically dependent over K(u). We therefore have an equation F (u; λ0, . . . , λd ) = 0. We note that Pn if (ξ ′ ) is a conjugate of (ξ) then λ′0 = − u0j ξj′ is a conjugate of λ0 and (ξ) 6= (ξ ′) if j=0 and only if λ0 6= λ′0 . Bertini’s theorem proves the following lemma: Lemma 3.3. Let V ⊂ Pn be an irreducible variety of dimension d ≥ 1 and Sn−d u a generic u linear subspace of dimension n − d. The intersection V ∩ Sn−d is an irreducible variety of dimension 0 over K(u1 , . . . , ud ) or the ideal (pK(u1 , . . . , ud)[x], ℓ1u , . . . , ℓdu ) = Pu is a 0-dimensional prime ideal of K(u1 , . . . , ud )[x]. The varieties of dimension 0 over a field consist of a finite number of points. We shall denote by T = {qi = (λi0 , λi1 , . . . , λin )|i = 1, 2, . . . , s} the intersection V ∩ H1u ∩ 17
Dam Van Nhi . . . ∩Hdu . All qi are conjugate to each other over K(u) := K(u1 , . . . , ud ). Over K(u)[u0 ], we define F (u0, u) to be the form s Y (u00 λi0 + u01 λi1 + · · · + u0n λin ). i=1 This form F (u0 , u) is the ground form of V and is irreducible over K(u)[u0 ]. We now consider V as an algebraic set over K. We have a decomposition of V into Sr K-varieties Vi and V = Vi . Hence, over K(u) we obtain i=1 r [ V ∩ H1u ∩ . . . ∩ Hdu = [Vi ∩ H1u ∩ . . . ∩ Hdu ]. i=1 Denote the ground form of Vi ∩ H1u ∩ . . . ∩ Hdu by Fi (u0, u) with i = 1, 2, . . . , r. Q r Then we have F (u0, u) = Fi (u0 , u) and each Fi (u0, u) is an irreducible form. i=1 Lemma 3.4. The ground form Fi (u0 , u) is absolutely irreducible and there are isomorphisms φij (Fi (u0 , u)) = Fj (u0 , u) for all i, j = 1, . . . , r. Proof. Denote Vi ∩ H1u ∩ . . . ∩ Hdu by Vi∗ . By Bertini’s theorem, every variety Vi∗ is irreducible. Since K is algebraically closed and all uij ∈ Ω, therefore K(u) is also algebraically closed in Ω by [9, lemma 4]. Hence Vi∗ and it’s ground form Fi (u0 , u) is absolutely irreducible. Suppose that qi and qj are generic points of Vi∗ and Vj∗ , respectively. It is obvious that all of qi are generic points of the irreducible variety V ∩ H1u ∩ . . . ∩ Hdu over K(u). By [5, I.5 Theorem 9], there is an isomorphism ϕij such that ϕij (qi ) = qj . Extend this isomorphism ϕij to an isomorphism δij of K(u)(qi ) onto K(u)(qj ), whose restriction to K(u) is an isomorphism, but not necessarily identity. The isomorphism δij induce an isomorphism φij such that φij (Fi (u0 , u)) = Fj (u0, u) for all i, j = 1, . . . , r. α S r Consider the intersection V ∩ Sn−d . Then, we have V ∩ H1α ∩ . . . ∩ Hdα = [Vi ∩ i=1 H1α ∩ . . . ∩ Hdα ] and the ground form of V ∩ H1α ∩ . . . ∩ Hdα is F (u0, α), and the ground form of Vi ∩ H1α ∩ . . . ∩ Hdα is Fi (u0, α), see [4]. Corollary 3.1. Every variety Vi ∩ H1α ∩ . . . ∩ Hdα and it’s ground form Fi (u0, α) are absolutely irreducible for almost all α. Proof. By lemma 3.2, the variety Vi ∩ H1α ∩ . . . ∩ Hdα is absolutely irreducible for almost all α. Because the ground form Fi (u0 , u) is absolutely irreducible by lemma 3.4, therefore Fi (u0 , α) is absolutely irreducible for almost all α by lemma 2.3. 18
The linear subspace section of variety by specializations The following result follows from lemma 2.4, lemma 3.4 and corollary 2.2: Theorem 3.1. For almost all α, there are isomorphisms Φij (Fi (u0 , α)) = Fj (u0 , α), ∀i, j = 1, . . . , r. REFERENCES ¨ [1] W. Gr¨obner, 1970. Uber das Reduzibilit¨atsideal eines Polynoms. Journal f¨ur die reine und angewandte Mathematik 239/240, 214-219. [2] J. Harris, 1980. The Genus of Space Curves. Math. Ann. 249, pp. 191-204. [3] W. V. D. Hodge and D. Pedoe, 1953. Methods of algebraic geometry, vol. II. The Syndics of the Cambridge University Press. [4] W. Krull, 1949. Parameterspezialisierung in Polynomringen II, Grundpolynom. Arch. Math. 1, pp. 129-137. [5] S. Lang, 1964. Introduction to algebraic geometry. New York: Interscience. [6] D.V. Nhi and N.V. Trung, 1999. Specialization of modules. Comm. Algebra 27, pp. 2959-2978. [7] D. V. Nhi, 2002. Specialization of graded modules. Proc. Edinburgh Math. Soc. 45, pp. 491-506. [8] E. Noether, 1923. Eliminationstheorie und allgemeine Idealtheorie. Math. Ann. vol. 90, pp. 229-261. [9] A. Seidenberg, 1950. The hyperplane sections of normal varieties. Tran. Amer. Math. Soc. 69, pp. 375-386. [10] N. V. Trung, 1980. Spezialisierungen allgemeiner Hyperfl¨achenschnitte und Anwendungen, in: Seminar D.Eisenbud/B.Singh/W.Vogel, vol. 1, Teubner-Texte zur Mathematik, Band 29, 4-43. [11] B.L. van der Waerden, 1958. Zur algebraischen Geometrie 19. Grundpolynom und zugeordnete Form. Math. Annalen, Bd 136, S., 139-155. 19