Variety of birational maps of degree d of Pn

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A natural and simple question asked is: Does the Cremona group Cr(n) admit a structure that is of an algebraic group of infinite dimension. This is still an open question because we don’t know if the set Cr≤d(n) of birational maps of degree ≤ d admits a structure of the algebraic variety.

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JOURNAL OF SCIENCE OF HNUE Mathematical and Physical Sci., 2013, Vol. 58, No. 7, pp. 50-58 This paper is available online at http://stdb.hnue.edu.vn VARIETY OF BIRATIONAL MAPS OF DEGREE d of Pn k Nguyen Dat Dang Faculty of Mathematics, Hanoi National University of Education Abstract. Let Sd = k[x0 , . . . , xn ]d be the k-vector space of homogeneous polynomials of degree d in (n + 1)-variables x0 , . . . , xn and the zero polynomial over an algebraically closed field k of characteristic 0. In this paper, we show that the birational maps of degree d of the projective space Pnk form a locally closed subvariety of the projective space P(Sdn+1 ) associated with Sdn+1 , denoted Crd (n). We also prove the existence of the quotient variety PGL(n + 1)Crd (n) that parametrize all the birational maps of degree d of P(Sdn+1 ) modulo the projective linear group PGL(n + 1) on the left. Keywords: Birational map, Cremona group, Grassmannian. 1. Introduction Let Cr(n) = Bir(Pnk ) denote the set of all birational maps of projective space Pnk . 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 for the first time by Luigi Cremona (1830 - 1903), an Italian mathematician. Although it has been studied since the 19th centery 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. The first important result is the theorem of Max Noether (1871): The Cremona group Bir(P2C ) of the complex projective plane P2C is generated by its subgroup PGL(3) and the standard quadratic transformation ω = [x0 x1 : x1 x2 : x2 x0 ], as an abstract group. This theorem was proved completely by Castelnuovo in 1901. This statement is only true if the dimension n = 2. The case n > 2, Ivan Pan proved a result following Hudson’s work on the generation of the Cremona group (see [6]). Received September 10, 2013. Accepted October 30, 2013. Contact Nguyen Dat Dang, e-mail address: dangnd@hnue.edu.vn 50
Variety of birational maps of degree d of Pnk One of the approachs in the study of the Cremona group is based on the knowledge of its subgroups. These studies were started by Bertini, Kantor and Wiman in the 1890s. Many important results have stemmed from this approach. For example, in 1893, Enriques determined the maximal connected algebraic subgroups of Bir(P2C ). In 1970, Demazure classified all the algebraic subgroups of rank maximal of Cr(n) with the aid of Enriques systems (see [2]). More recently, in 2000, Beauville and Bayle gave the classification of birational involutions up to birational conjugation. And then, in 2006, Blanc, Dolgachev and Iskovskikh also gave the classification of finite subgroups of Bir(P2C ) (see [1]). In 2009, Serge Cantat showed that Bir(P2C ) is simple as an abstract group. In the 1970s, Shafarevich published an article (see [8]) with the title: On some infinite dimensional groups, in which he showed that the group G = Aut(AnC ) of all polynomial automorphisms of the affine space AnC admits a structure of an algebraic group of infinite dimension with the natural filtration: G = ∪∞ d=1 G≤d where G≤d is the affine algebraic variety of all the polynomial automorphisms of degree ≤ d of the affine space AnC . He also calculated its Lie algebra. So, he proved that the group Aut(AnC ) is not simple as an abstract group. A natural and simple question asked is: Does the Cremona group Cr(n) admit a structure that is of an algebraic group of infinite dimension. This is still an open question because we don’t know if the set Cr≤d (n) of birational maps of degree ≤ d admits a structure of the algebraic variety. However, the answer is "yes" for the set Crd (n) of all birational maps of degree d of the projective space Pnk and PGL(n + 1)Crd (n). These results are related to my PhD thesis (see [5]) that was successfully defended in 2009 at the Université de Nice (in France) but which has not yet been published in any journal. 2. Subvariety Crd (n) ⊂ P(Sdn+1 ) In classic algebraic geometry, we know that a rational map of the projective space Pnk is of the form: [ ] Pnk ∋ [x0 : . . . : xn ] = x 99K φ(x) = P0 (x) : . . . : Pn (x) ∈ Pnk , 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 deg φ. In the language of linear systems; giving a rational map such as φ is equivalent to giving a linear system without fixed components of Pnk { n } ∑ φ⋆ |OPn (1)| = λi Pi |λi ∈ k . i=0 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)|. 51
Nguyen Dat Dang Note that a rational map φ : Pnk 99K Pnk is not in general a map of the set Pnk to Pnk ; it is only the map defined in its domain of definition Dom(φ) = Pnk \V (P0 , . . . , Pn ). We say that φ is dominant if its image φ(Dom(φ)) is dense in Pnk . By the Chevalley theorem, the image φ(Dom(φ)) is always a constructible subset of Pnk , hence, it is dense in Pnk if and only if it contains a non-empty Zariski open subset of Pnk (see page 94, [4]). 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 the dominant rational maps φ : Pnk 99K Pnk is identified with the set of injective field homomorphisms φ⋆ of the field of all the rational fractions k(x1 , . . . , xn ) in n-variables x1 , . . . , xn . We say that a rational map φ : Pnk 99K Pnk is birational (a birational automorphism) if there exists a rational map ψ : Pnk 99K Pnk 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 dominant. If we denote by Cr(n) = Bir(Pnk ) the set of all birational maps of the projective space Pnk , then Cr(n) is a group under composition of dominant rational maps and 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 . We immediately have the two following propositions: Proposition 2.1. A rational map φ : Pnk 99K Pnk is birational if and only if it is dominant and there exists a rational map ψ : Pnk 99K Pnk such that ψ ◦ φ = idPn . Proof of Proposition 2.1. The necessary condition is obvious. Conversely, by assumption, we have an injective field homomorphism φ⋆ : k(x1 , . . . , xn ) → k(x1 , . . . , xn ). If there exists a rational map ψ : Pnk 99K Pnk verifying ψ ◦ φ = idPn , then ψ is dominant. Hence ψ ⋆ : k(x1 , . . . , xn ) → k(x1 , . . . , xn ) is also injective. Moreover, we have: ( )⋆ φ⋆ ◦ ψ ⋆ = (ψ ◦ φ)⋆ = idPn = idk(x1 ,...,xn ) . Consequently, φ⋆ is also surjective, hence an automorphism. In other words, φ is birational. Proposition 2.2. If φ : Pnk 99K Pnk is a birational map, then deg φ−1 6 (deg φ)n−1 . Proof of Proposition 2.2. Denote X the locus of undefined points of φ−1 . If Z is a generic linear subvariety of Pnk , we denote Ze := φ−1 (Z r X) the Zariski closure, and we call it the strict transform of Z by φ. By definition, the degree of φ is also the degree of the strict transform of a generic hyperplane H ∈ |OPn (1)|, that is, deg φ = deg H.e We will −1 show that the degree of φ is equal to the degree of the strict transform of a generic line: deg φ−1 = deg L. e Indeed, consider the subvariety of incident lines { ( ) } C1 (X) = L ∈ G 1, Pn |L ∩ X ̸= ∅ 52
Variety of birational maps of degree d of Pnk ( ) where G 1, Pn is the grassmannian of all the lines in Pn . We have ( ) dim C1 (X) = 1.(n − 1) + dim X < 2(n − 2) = dim G 1, Pn . ( ) Hence, C1 (X) $ G 1, Pn . Consequently, there exists a generic line L ⊂ Pn such that L ∩ X = ∅. The restriction of φ−1 to L is described by a linear system without base points and deg φ−1 = deg L.e Since L is a generic line, we can write: L = H1 ∩ · · · ∩ Hn−1 as the complete intersection of the generic hyperplanes of Pn . Therefore ∩ n−1 ( ) ( ) deg φ −1 e = deg = deg L e i 6 deg H H e i n−1 = deg φ n−1 . i=1 If V is a k-vector space, we denote by P(V ) = (V − {0})/k∗ the projective space associated with V , whose points are one-dimensional vector subspaces of V . In particular, when V = Sd(n+1 is ) the k-vector space of (n + 1)-uples (n+d) of d-forms in (n + 1)-variables, dim Sd = d .(n + 1), then[ dim P(Sd ) ]= d ( .(n + n+1 n+d n+1 ) 1) − 1. The homogeneous coordinates of each point φ = P0 : . . . : Pn ∈ P Sd n+1 are the coefficients of the polynomials P0 , . . . , Pn . Now, we present the most important result of this section: Theorem 2.3. The set Crd (n) of all birational maps of degree d of the projective space Pnk is a locally closed subvariety of the projective space P(Sdn+1 ). In order to prove Theorem 2.3, we need the following lemmas: Lemma 2.4. A rational map φ : Pnk 99K Pnk is dominant if and only if its jacobian determinant is not zero. Proof of Lemma 2.4. The necessary condition: If φ is a dominant rational map, then φ : Dom(φ) → Pn is a dominant morphism of integral schemes of finite type over k. According to Proposition 10.4, in [4], page 270-273, there is a nonempty open subset U ⊂ Dom(φ) ⊂ Pn such that φ : U → Pn is a smooth morphism of relative dimension dim(U ) − n = 0, that is, an étale morphism. By definition, its tangent linear map ∼ Tx φ : Tx U → Tx Pn is an isomorphism of vector spaces, for all x. Hence, its determinant, which is also the jacobian determinant of φ must be not zero: Jac(φ)(x) = det(Tx φ) ̸= 0, for all x ∈ U. [ ] The sufficient condition: If φ(x) = P0 (x) : . . . : Pn (x) is ( a non-dominant ) rational ∂Pi map, we will prove that its jacobian determinant Jac(φ) = det ≡ 0. Indeed, since ∂xj φ(Dom(φ)) ⊂ φ(Dom(φ)) and the Zariski closure φ(Dom(φ)) is a proper closed subset of Pnk , hence, the image of φ must be contained in some hypersurface F (x0 , . . . , xn ) = 0, that is, F (P0 (x), . . . , Pn (x)) = 0. We can suppose F is a homogeneous polynomial of 53
Nguyen Dat Dang least degree such that F (P0 (x), . . . , Pn (x)) = 0. Hence, we have for all x   ∂F (P0 (x), . . . , Pn (x)) ∑ n ∂F ∂Pi   0 = = (P0 (x), . . . , Pn (x)) (x)   ∂x0 i=0 ∂xi ∂x0   ... ...   ... ...   ∑   ∂F (P 0 (x), . . . , Pn (x)) n ∂F ∂Pi  0= = (P0 (x), . . . , Pn (x)) (x). ∂xn i=0 ∂xi ∂xn ∂F Since deg( ) < deg F for all i and by the hypothesis of the smallest degree of ∂xi ∂F F , the partial derivatives (P0 (x), . . . , Pn (x)) must be non-zero. Consequently, the ( ∂xi ) ∂P0 ∂Pn vectors vi = (x), . . . , (x) , i = 0, . . . , n of the k(x0 , . . . , xn )-vector space ∂xi ∂xi k(x0 , . . . , xn )n+1 are linearly dependent. Consequently, the determinant of the family of these vectors must be zero. In other words, the jacobian determinant Jac(φ) = 0. { [ ] ( ) } Corollary 2.5. Ud = φ = P0 : . . . : Pn ∈( P Sdn+1 ) | φ : P n 99K P n dominant is a Zariski open subset of the projective space P Sd n+1 . ( ) Proof of Corollary 2.5. This is obvious because Ud = P Sdn+1 \ V (Jac)( is the ) complement of the projective algebraic set V (Jac) ( n+1 ) in the projective space P S n+1 d . Here V (Jac) is the projective algebraic set in P Sd defined by the annulation of all coefficients of the polynomial Jac. { [ ] ( ) } Lemma 2.6. Vd = φ = P0 : . . . : Pn ∈ (P Sdn+1 ) | P i are mutually prime is also a Zariski open subset of the projective space P Sd n+1 . Proof of Lemma 2.6. We consider the following regular map (a variant of the Segre embedding) ( n+1 ) ( ) ( ) sd,r : P Sd−r × P Sr −→ P Sdn+1 ([ ] ) [ ] P0 : . . . : Pn ; P 7−→ P0 P : . . . : Pn P . According to Theorem ( n+13.13, ) page 38, in [3], the image [ of this regular ] map is a) Zariski ( n+1 closed subset of P Sd , corresponding to the points Q0 : . . . : Qn (∈ P S )d having a common divisor of degree r. Hence, the complement Vd,r := P Sdn+1 \Imsd,r is a ( ) ∩ d−1 Zariski open subset of P Sdn+1 . Clearly, the intersection Vd = Vd,r is also open. r=1 { [ ] ( ) ( ) } Lemma 2.7. Fd = φ = P0 : . . . : Pn ∈ P (Sdn+1 )| ∃ ψ ∈ P Sen+1 , ψ ◦ φ = idPn is a Zariski closed subset of the projective space P Sdn+1 , where we denote e = dn−1 . 54
Variety of birational maps of degree d of Pnk Proof of Lemma 2.7. We consider the following regular map (the first projection): ( ) ( ) ( ) pd,e : P Sdn+1 × P Sen+1 −→ P Sdn+1 where e = dn−1 ([ ] [ ]) [ ] P0 : . . . : Pn ; Q0 : . . . : Qn 7−→ P0 : . . . : Pn . ∑ ([ ] [ ]) ( n+1 ) The ( n+1 ) set of points (φ, ψ) = P : . . . : P ; Q : . . . : Q ∈ P Sd × d,e 0 n ∧ 0 n P Se such ( that) ψ ◦(φ = )idP , that is, (ψ ◦ φ)(x) x = 0, ∀ x is a Zariski closed ∑ n subset of P Sdn+1 × P Sen+1 . It is easy to find that Fd is also the image of d,e by pd,e , and that it is a Zariski closed subset of P(Sdn+1 ) by Theorem 3.12, page 38, in [3]. Proof of Theorem 2.3. According to Proposition 2.1, we have:    [ ] ( ) (i) φ : P n 99K P n is dominant  Crd (n) = φ = P0 : . . . : Pn ∈ P Sdn+1 (ii) the Pi are( mutually ) prime   (iii) ∃ ψ ∈ P Sen+1 , ψ ◦ φ = idPn where e is chosen equal to dn−1 because if φ is a birational map of degree d, the deg φ−1 ≤ dn−1 by Proposition 2.2. Hence, Crd (n) is the intersection Crd (n) = Ud ∩ Vd ∩ Fd of the open set Ud ∩ Vd (by Corollary 2.5 and Lemma 2.6)( and of ) the closed set Fd (by Lemma 2.7). Consequently, it is a locally closed subset of P Sdn+1 . ( ) 3. Subvariety PGL(n + 1)Crd (n) in G n + 1, Sd While the projective linear group PGL(n + 1) is obviously an algebraic group of the Cremona group Cr(n), it is not normal in Cr(n). Hence, we obtain the two distinct quotient sets: PGL(n + 1)Cr(n) = {PGL(n + 1) ◦ φ : φ ∈ Cr(n)} , Cr(n)PGL(n + 1) = {φ ◦ PGL(n + 1) : φ ∈ Cr(n)} . Here, we will not speak of them and we will study only the subset PGL(n + 1)Crd (n) of the first PGL(n + 1)Crd (n) = {PGL(n + 1) ◦ φ : φ ∈ Crd (n)} . From the viewpoint of algebraic geometry, we know that the algebraic group PGL(n + 1) acts on the variety Crd (n) by the left multiplication PGL(n + 1) × Crd (n) −→ Crd (n) (u, φ) 7−→ u ◦ φ. 55
Nguyen Dat Dang We have a natural question: Does the set PGL(n + 1)Crd (n) of all the distinct orbits of Crd (n) by the action of PGL(n + 1) admit the structure of quotient variety? In order to answer this question, we need recall that the k-planes of a given vector( space ) V form an algebraic variety, called the grassmannian ( )of k-planes of V , denoted G k, V . In particular, we have ( the grassmannian ) G n + 1, S d of (n + ( 1)-planes of ) the vector space Sd . Evidently, G n + 1, Sd is also the grassmannian G n, |OPn (d)| of all the linear subvariety of dimension n of the linear system |OPn (d)|. Theorem 3.1. The set PGL(n + 1)Crd (n) of all the distinct orbits of Crd (n) by the action ) + 1) can be identified as a locally closed subvariety of the grassmannian ( of PGL(n G n + 1, Sd of (n + 1)-planes of the vector space Sd . In order to prove this theorem, we need the following lemma: Lemma 3.2. We have the two following results: [ ] (i) If φ = P0 : . . . : Pn is a birational map of degree d, then the vectors P0 , . . . , Pn are linearly independent in Sd . {[ ] ( ) } (ii) The set Ud (n) = P0 : . .(. : Pn ) ∈ P Sdn+1 : The Pi are linearly independent is a Zariski open subset of P Sdn+1 . Proof of Lemma 3.2. (i). If they were linearly dependent in Sd , without loss of generality, we could suppose: P0 = λ1 P1 + · · · + λn Pn with λi ∈ k. Hence, the image of φ would be contained in the hyperplane x0 − λ1 x1 − · · · − λn xn = 0, so that it would not be dense in Pnk . Therefore, φ would not be birational. (ii). The Pi are linearly independent in Sd if and only if the rank of the (n + 1) × dim(Sd )-matrix (P0 . . . Pn ) formed by the coefficients [ ] the P of all is equal ( i n+1 ) to n + 1. If we denote Fd (n), the set of all the φ = P0 : . . . : Pn ∈ P Sd such that the Pi are linearly dependent in Sd , that is, rank(P0 . . . Pn ) < n + 1, then Fd (n) is a closed ( n+1 ) of all the (n + 1)-sub-determinant of the matrix subvariety defined by the annulation (P0 . . . Pn ). Hence, Ud (n) = P Sd − Fd (n) is a Zariski open subset. Proof of Theorem 3.1. On the Zariski open set Ud (n), we have a natural surjective map: ( ) Φn,d : Ud (n) G n + 1, Sd [ ] φ = P0 : . . . : Pn −7 → Span(P0 , . . . , Pn ) { } ∑ n where Span(P0 , . . . , Pn ) = λi Pi : λi ∈ k is the k-vector space spanned by i=0 P0 , . . . , Pn . This map ( is also a surjective ) morphism of schemes. Moreover, the morphism Φn,d : Ud (n) G n + 1, Sd is still the principal bundle with the fiber PGL(n + 1) 56
Variety of birational maps of degree d of Pnk ( ) and the structural group PGL(n + 1). Therefore, the grassmannian G n + 1, Sd ∼ = PGL(n + 1)Ud (n) is the quotient of the space Ud (n) by PGL(n + 1). ( ) If we denote Gd (n) = Φn,d Crd (n) the image of the locally closed subvariety Crd (n) by Φn,d , then by the surjectivity of Φn,d , we obtain: Crd (n) = (Φn,d )−1 (Gd (n)) . −1 ( n+1 ) to Theorem 2.3, (Φn,d ) (Gd (n)) = Crd (n) is a locally closed subvariety of According P Sd , and also of Ud (n). By the property of(the principal ) bundle, Gd (n) is also a locally closed subvariety of the grassmannian G n + 1, Sd . Hence, the restriction to Crd (n) of Φn,d gives us a surjective morphism of schemes, also denoted Φn,d ( ) Φn,d : Crd (n) Gd (n) ⊂ G n + 1, Sd [ ] φ = P0 : . . . : Pn 7−→ Span(φ). Then, we obtain the cartesian square Φn,d ( ) Ud (n) / G n + 1, Sd O O inclusion inclusion Φn,d Crd (n) / Gd (n) Therefore, Φn,d : Crd (n) Gd (n) is also a principal bundle with the fiber PGL(n + 1) and the structural group PGL(n + 1). Consequently, Gd (n) ∼= PGL(n + 1)Crd (n) is the quotient of the variety Crd (n) by PGL(n + 1). In summary, we have an isomorphism: ∼ ( ) PGL(n + 1)Crd (n) −→ Gd (n) ⊂ G n + 1, Sd PGL(n + 1) ◦ φ 7−→ Span(φ). 4. Conclusion In this paper, the author has proven two main results. The first is Theorem 2.3: The set Crd (n) of birational maps of degree d of the projective space Pnk is a locally closed subvariety of the projective space P(Sdn+1 ). The second is Theorem 3.1, which proves the existence of the quotient variety PGL(n + 1)Crd (n) that parametrize all birational maps of degree d of P(Sdn+1 ) modulo the projective linear group PGL(n + 1) on the left. In the next publications, the author will continue to give new results on the irreductible components of the quotient variety PGL(n + 1)Crd (n). 57
Nguyen Dat Dang REFERENCES [1] Jérémy Blanc, 2006. Thèse: Finite abelian subgroups of the Cremona group of the plane. En ligne: http://www.unige.ch/cyberdocuments/theses2006/BlancJ/these.pdf. [2] Michel Demazure, 1970. Sous-groupes algébriques de rang maximum du groupe de Cremona. Ann. scient. Éc. Norm. Sup., 4e série, t. 3, p. 507 à 588. En ligne. http://archive.numdam.org. [3] Joe Harris, 1992. Algebraic Geometry. Graduate Texts in Mathematics, Springer Verlag. [4] Robin Hartshorne, 1977. Algebraic Geometry. New York Heidelberg Berlin. Springer Verlag. [5] Dat Dang Nguyen, 2009. Groupe de Cremona. Thèse in Université de Nice, in France. [6] Ivan Pan, 1999. Une remarque sur la génération du groupe de Cremona. Sociedade Brasileira de Matemática, Volume 30, Issue 1, pp 95-98. [7] Ivan Pan and Alvaro Rittatore, 2012. Some remarks about the Zariski topology of the Cremona group. Online on http://www.cmat.edu.uy/ ivan/preprints/cremona130218.pdf [8] I.R. Shafarevich, 1982. On some infinitedimensional groups. American Mathematical Society. 58