Extended degree of rational maps

HNUE JOURNAL OF SCIENCE DOI: 10.18173/2354-1059.2019-0027 Natural Science, 2019, Volume 64, Issue 6, pp. 23-30 This paper is available online at EXTENDED DEGREE OF RATIONAL MAPS Nguyen Dat Dang Faculty of Mathematics, Hanoi National University of Education Abstract. In the article [1], we see that the birational maps of degree d of the projective space Pn k form a locally closed subvariety of the projective space P(Sn+1d ), denoted Crd(n). In this paper, we will construct the notion of extended degree of rational maps, then we obtain the functor crd(n) that represents the variety Crd(n). Keywords: Rational map, birational map, extended degree, representable functor, remona group. 1. Introduction Let Cr(n) = Bir(Pn k ) denote the set of all birational maps of the projective space P n k , on the field k. It is clear that Cr(n) is a group under composition of dominant rational maps called the Cremona group of order n. This group is naturally identified with the Galois group of k-automorphisms of the field k(x1, . . . , xn) of rational fractions in n-variables x1, . . . , xn. It was studied in the first time by Luigi Cremona (1830 - 1903), an Italian mathematician. Although it has been studied since the 19th century by many famous mathematicians, it is still not well understood. For example, we still don’t know if it has the structure of an algebraic group of infinite dimension. In the article [2], we constructed the Cremona group k-functor cr(n) : Alg(k) −→ Gr R 7−→ BirR(PnR). from the category Alg(k) of k-algebras to the category Gr of groups and calculated its Lie algebra. The k-value points of the Cremona group k-functor cr(n) are exactly the elements of the Cremona group Crk(n) = Birk(Pnk). Received February 25, 2019. Revised June 21, 2019. Accepted June 28, 2019. Contact Nguyen Dat Dang, e-mail address: dangnd@hnue.edu.vn 23 Nguyen Dat Dang Denote by Sd = k[x0, . . . , xn]d the k-vector space of homogeneous polynomials of degree d in (n + 1)-variables x0, . . . , xn and zero polynomial over an algebraically closed field k of characteristic 0. In the article [1], we also knew that the birational maps of degree d of the projective space Pn k form a locally closed subvariety of the projective space P(Sn+1d ) associated with the k-vector space Sn+1d , denoted Crd(n). In this paper, we will construct the sub-functor crd(n) of the Cremona group k-functor cr(n) and we will show that this sub-functor crd(n) represents the variety Crd(n). For that, we need extend the notion of degree of rational maps ϕ : PnR 99K PnR where R is not necessarily a field but any k-algebra. 2. Content 2.1. Degrees of rational maps In classic algebraic geometry, we know that a rational map of the projective space P n k is of the form: P n k ∋ [x0 : . . . : xn] = x 99K ϕ(x) = [ P0(x) : . . . : Pn(x) ] ∈ Pn k , where P0, . . . , Pn are homogeneous polynomials of same degree in (n + 1)-variables x0, . . . , xn and are mutually prime. The common degree of Pi is called the degree of ϕ; denoted by degϕ. In the language of linear systems; giving a rational map such as ϕ is equivalent to giving a linear system without fixed components of Pn k ϕ⋆|OPn(1)| = { n∑ i=0 λiPi|λi ∈ k } . Clearly, the degree of ϕ is also the degree of a generic element of ϕ⋆|OPn(1)| and the undefined points of ϕ are exactly the base points of ϕ⋆|OPn(1)|. Note that a rational map ϕ : Pn k 99K Pn k is not in general a map from the set Pn k to P n k ; it is only the map defined on its domain of definition Dom(ϕ) = Pn k \ V (P0, . . . , Pn). We say that ϕ is dominant if its image ϕ(Dom(ϕ)) is dense in Pn k . By the Chevalley theorem, the image ϕ(Dom(ϕ)) is always a constructible subset of Pn k , hence, it is dense in Pn k if and only if it contains a non-empty Zariski open subset of Pn k (see the page 94, in [3]). In general, we can not compose two rational maps. However, the composition ψ ◦ ϕ is always defined if ϕ is dominant so that the set of all dominant rational maps ϕ : Pn k 99K Pn k is identified with the set of injective field homomorphisms ϕ⋆ of the field k(x1, . . . , xn) of all rational fractions in n-variables x1, . . . , xn. We say that a rational map ϕ : Pn k 99K Pn k is birational (a birational automorphism) if there exists a rational map ψ : Pn k 99K Pn k such that ψ ◦ ϕ = idPn = ϕ ◦ ψ as rational maps. Clearly, if such a ψ exists, then it is unique and is called the inverse of ϕ. Moreover, ϕ and ψ are both 24 Extended degree of rational maps dominant. If we denote by Cr(n) = Bir(Pn k ) the set of all birational maps of the projective space Pn k , then Cr(n) is a group under composition of dominant rational maps an is called the Cremona group of order n. This group is naturally identified with the Galois group of k-automorphisms of the field k(x1, . . . , xn) of rational fractions in n-variables x1, . . . , xn. Now, suppose that R is any k-algebra and ϕ : PnR 99K PnR is a rational map (see the general definition of rational maps in [4]). It gives us a family of rational κ(s)-maps( ϕs : P n κ(s) 99K P n κ(s) ) s∈Spec(R) where κ(s) is the residue field at s. Roughly, we say that the degree of ϕ is equal to d if ϕs has the degree d for all s ∈ Spec(R). More precisely, we define as follows: Definition 2.1. Let X = Spec(R) be an affine k-scheme and ϕ : PnR 99K PnR a rational R-map with the domain of definition U = dom(ϕ) ⊂ PnR satisfying the following condition: codim ( P n κ(s) − Us,P n κ(s) ) > 2, ∀s ∈ Spec(R) (2.1) where Us := U ×Spec(R) Specκ(s) is the fiber of U at s. Suppose there exists an invertible sheaf L on Spec(R) and a positive integer d such that ϕ⋆OPn R (1) ≃ OU(d)⊗OU p ⋆ L (2.2) where OU (d) is the restriction to U of OPn R (d) and p is the restriction of the structural morphism pi to U , and here ϕ : U → PnR becomes a morphism, in such a way that, the notation ϕ⋆OPn R (1) defines an invertible sheaf on U . It is clear that such a couple (L , d) is uniquely determined if it exists. This positive integer d is called the degree of ϕ, still denoted by deg(ϕ). Remark 2.1. When R = k is any field, X = Spec(k) and pi : Pn k → Spec(k) is the structural morphism, we show that such a couple (L , d) always exists and it is uniquely determined, so we find again the notion of usual degree in classical algebraic geometry. Indeed, the invertible sheaf L is quasi-coherent on X = Spec(k), therefore, of form L = M˜ where M is a locally free k-module of rank 1 (projective of rank 1), hence, M ≃ k, that is, L = k˜ = OSpec(k). Therefore, p⋆L ≃ p⋆OSpec(k) ≃ OU , hence, the isomorphism of sheaves (2.2) becomes: ϕ⋆OPn k (1) ≃ OU(d). Moreover, in this case, (2.1) gives us Pic(U) = Pic(Pn k ), hence, |OPn k (d)| ≡ |OU(d)| ⊃ ϕ ⋆|OPn k (1)|, therefore, the 25 Nguyen Dat Dang generic element (generic hypersurface) of the linear system defining ϕ has for the degree d, ie, the degree of ϕ is d, as the usual notion of degree. Remark 2.2. The condition (2.2) implicates that the degree of ϕs is equal to d for all s ∈ Spec(R). Indeed, by taking the fiber at s ∈ Spec(R), the isomorphism of sheaves (2.2) becomes ϕ⋆sOPnκ(s)(1) ≃ OUs(d) = OP n κ(s) (d)|Us. Therefore, the linear system Cϕs := ϕ⋆s|OPnκ(s)(1)| ⊂ |OUs(d)| ≡ |OPnκ(s)(d)| defining ϕs is a linear system of hypersurfaces of degree d in Pnκ(s), hence, deg(ϕs) = d. Remark 2.3. We consider the following example ϕ : P2 A1 k 99K P2 A1 k , ϕ = [xz : y(z + sx) : z2] where the parameter s ∈ A1 k = X = Speck[t], we have codim ( P 2 κ(s) − Us,P 2 κ(s) ) = 2, deg(ϕs) = 2, ∀s 6= 0 codim ( P 2 κ(0) − U0,P 2 κ(0) ) = 1, deg(ϕ0) = 1 6= 2. Then, this rational map ϕ has not degree. This example also shows us that the condition (2.1), that gives us: Pic(Us) = Pic(Pnκ(s)), ∀s ∈ Spec(R), is necessary. 2.2. Main results We consider the Cremona group k-functor cr(n) : Alg(k) −→ Gr R 7−→ BirR(PnR) from the category Alg(k) of k-algebras to the category Gr of groups. Here, we denote by BirR(PnR) the group of birational R-maps of the projective space PnR on R. According to Definition 2.1, we can denote by Bird,R ( P n R ) the set of birational R-maps of degree d of the projective space PnR on the k-algebra R. Therefore, we obtain the sub-functor following of the Cremona group k-functor cr(n) crd(n) : Alg(k) −→ Set R 7−→ crd(n)(R) := Bird,R ( P n R ) . Here, we denote by Set the category of sets. We can also regard crd(n) as a k-functor defined on the category Sch(k) of k-schemes. Now, we obtain the main result following: Theorem 2.1. The restriction of the k-functor crd(n) to the category of noetherian k-schemes is a k-functor representable by the scheme Crd(n). 26 Extended degree of rational maps Proof. It suffices to establish an isomorphism of k-functors crd(n) ≃ hCrd(n) with the representable functor hCrd(n), that is, for all noetherian k-algebra R, a bijection crd(n)(R) = Bird,R ( P n R ) ∼ −→ MorSch(k) (Spec(R),Crd(n)) = hCrd(n) ( Spec(R) ) and for all homomorphism of noetherian k-algebras T → R, a commutative square Bird,R ( P n R ) ∼ −→ MorSch(k) (SpecR,Crd(n)) ↑ ↑ Bird,T ( P n T ) ∼ −→ MorSch(k) (SpecT,Crd(n)) . Let ϕ ∈ Bird,R ( P n R ) be a birational R-map of degree d of the projective space PnR on some noetherian k-algebra R. We try to construct a morphism of k-schemes ϕ′ : SpecR → Crd(n) ⊂ P ( Sn+1d ) , that is, a morphism into the projective space ϕ′ : SpecR → P(Sn+1d ), whose image is contained in Crd(n). Such a morphism will be defined by the data containing an invertible R-module H0 ( SpecR,L ) and an epimorphism of R-modules Sn+1d ⊗k R ։ H0 ( SpecR,L ) . According to the definition of degree (Definition 2.1), there exists always such an invertible sheaf L . In order to verify that L is suitable, we need prove some complementary results following: Lemma 2.1. Let A→ B be a local homomorphism of local noetherian rings such that B is a flat A-module. Then, if we denote by κ(A) the residue field of A, we have the equality following of depth: depth(B) = depth(A) + depth ( κ(A)⊗A B ) . Proof of Lemma 2.1 is also the corollary of Proposition 11, page AC X-13, in [5]. Lemma 2.2. Suppose that X is a locally noetherian scheme, Y is a closed subscheme of X and F is a coherent OX-module. Then, the following conditions are equivalent: (i) For all y ∈ Y , depth ( Fy ) ≥ 2. (ii) For all open subset V of X , the natural homomorphism following is bijective H0 ( V,F ) −→ H0 ( V ∩ (X − Y ),F ) . 27 Nguyen Dat Dang Proof of Lemma 2.2 is also Corollary III-1, page 11, in [6]. Lemma 2.3. Suppose that R is a noetherian k-algebra, ϕ : PnR ∼ 99K PnR is a birational R-map of degree d with U = dom(ϕ) ⊂ PnR the domain of definition of ϕ. Then, the canonical homomorphism of R-modules α : H0 ( P n R,OPnR(d) ) −→ H0 ( U,OU(d) ) x 7−→ x|U is an isomorphism. Proof. This is a direct consequence of Lemma 2.1 and Lemma 2.2. Indeed, we need only apply Lemma 2.2 with X = PnR, Y = PnR − U and F = OPnR(d). So the only problem remains verifying that for all y ∈ Y , depth ( Oy,Pn R (d) ) ≥ 2. However,OPn R (d) is an invertible sheaf, Oy,Pn R (d) ≃ Oy,Pn R . Now, by applying Lemma 2.1 to the canonical homomorphism A = Os,SpecR → B = Oy,Pn R where pi(y) = s. We have depth ( Oy,Pn R ) = depth (Os,SpecR) + depth ( κ(s)⊗Os,SpecR Oy,PnR ) = depth (Os,SpecR) + depth ( Oys,Pnκ(s) ) ≥ depth ( Oys,Pnκ(s) ) = dimKrull ( Oys,Pnκ(s) ) ≥ codim ( P n κ(s) − Us,P n κ(s) ) > 2. Here, we have the equality: depth ( Oys,Pnκ(s) ) = dimKrull ( Oys,Pnκ(s) ) because Pnκ(s) is smooth. Lemma 2.4. (Projection Formula) (see in [7]). Suppose that f : (X,OX) → (Y,OY ) is a morphism of ringed spaces, F is an OX -module and E is an OY -module locally free of finite rank. Then f⋆ ( F ⊗OX f ⋆ E ) ≃ f⋆ (F )⊗OY E . In fact, we have also the isomorphism Rif⋆ ( F ⊗OX f ⋆ E ) ≃ Rif⋆ (F )⊗OY E , ∀i. Lemma 2.5. Let R be a noetherian k-algebra, we have (a) H0 ( SpecR, pi⋆OPn R (d) ) ≃ Sd ⊗k R, (b) H0 ( SpecR, p⋆OU(d) ) ≃ Sd ⊗k R. 28 Extended degree of rational maps Proof. (a) Indeed: H0(SpecR, pi⋆OPn R (d) ) ≃ H0 ( P n R,OPnR(d) ) ≃ R [ x0, . . . , xn ] d ≃ Sd ⊗k R. (b) By applying Lemma 2.3, we have H0 ( SpecR, p⋆OU(d) ) ≃ H0 ( U,OU(d) ) ≃ H0 ( P n R,OPnR(d) ) ≃ R [ x0, . . . , xn ] d ≃ Sd⊗kR. Now, we come back to prove main Theorem 2.1. By applying the three previous lemmas and by taking p⋆ of two sides of the isomorphism (2.2), we have p⋆ϕ ⋆OPn R (1) ≃ p⋆ ( OU(d)⊗OU p ⋆ L ) ≃ p⋆ ( OU(d) ) ⊗OSpecR L ≃ Sd ⊗k L . According to the property of the adjoint functor, we alway have a morphism of sheaves OPn R (1)→ ϕ⋆ϕ ⋆OPn R (1) hence, a morphism of sheaves pi⋆OPn R (1) −→ pi⋆ϕ⋆ϕ ⋆OPn R (1) = p⋆ϕ ⋆OPn R (1) ≃ Sd ⊗k L . By taking the global sections, we have the following homomorphism: S1 ⊗k R −→ Sd ⊗k H 0 L . By taking the duality, we obtain Θ : (Sd) ⋆ ⊗k (S1 ⊗k R) −→ H 0 L . The monomials xI = xi00 · · ·xinn with Card(I) = i0 + · · ·+ in = d form a canonical basis of the vector k-space k[x0, . . . , xn]d = Sd, hence, we obtain the dual basis (xI)⋆ of (Sd)⋆ (xI) ⋆ : Sd −→ k xJ 7−→ (xI) ⋆ (xJ) := δI,J where δI,J is the Kronecker symbol. In the end, the homomorphism Θ is well defined by its images on the canonical basis (xI)⋆ ⊗ xi as follows: Θ : (Sd) ⋆ ⊗k (S1 ⊗k R) ։ H 0 L (xI) ⋆ ⊗ xi 7−→ Θ ( (xI) ⋆ ⊗ xi ) = CoeffxI (Pi) 29 Nguyen Dat Dang where CoeffxI (Pi) is the coefficient of xI in Pi if ϕ = [ P0 : . . . : Pn ] . This shows that Θ is an epimorphism of R-modules. In summary, we showed that the birational R-map ϕ = [ P0 : . . . : Pn ] of degree d determines an epimorphism of R-modules Sn+1d ⊗k R ։ H0L . So, this epimorphism defines a morphism of k-schemes ϕ′ : SpecR→ P ( Sn+1d ) . As the set map, the underlying mapping of the corresponding morphism ϕ′ is defined as follows: SpecR ∋ s 7→ ϕ′(s) = ϕs ∈ Crd(n) where ϕs is a member of the family of the birational k-maps ϕ. Therefore, the image of ϕ′ is contained in Crd(n). Conversely, all the morphisms of k-schemes from SpecR to Crd(n) are deduced from this way. Moreover, the fact that the square being commutative is obvious. 3. Conclusions In this paper, the author has acquired the two main results. The first is Definition 2.1 that is the notion of extended degree of rational maps. The second is Theorem 3.1, which states that the restriction of the k-functor crd(n) to the category of noetherian k-schemes is a k-functor representable by the scheme Crd(n). REFERENCES [1] Nguyen Dat Dang, 2013. 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