Variety of birational maps of degree d of Pn

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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Variety of birational maps of degree d of Pn 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; calledthe Cremona group of order n. This group is naturally identified with the Galois group ofk-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, itis still not well understood. For example, we still don’t know if it has the structure of analgebraic group of infinite dimension. The first important result is the theorem of Max Noether (1871): The Cremonagroup 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 abstractgroup. This theorem was proved completely by Castelnuovo in 1901. This statement isonly true if the dimension n = 2. The case n > 2, Ivan Pan proved a result followingHudson’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.vn50 Variety of birational maps of degree d of Pnk One of the approachs in the study of the Cremona group is based on the knowledgeof 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, Enriquesdetermined the maximal connected algebraic subgroups of Bir(P2C ). In 1970, Demazureclassified all the algebraic subgroups of rank maximal of Cr(n) with the aid of Enriquessystems (see [2]). More recently, in 2000, Beauville and Bayle gave the classification ofbirational involutions up to birational conjugation. And then, in 2006, Blanc, Dolgachevand Iskovskikh also gave the classification of finite subgroups of Bir(P2C ) (see [1]). In2009, 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 someinfinite dimensional groups, in which he showed that the group G = Aut(AnC ) of allpolynomial automorphisms of the affine space AnC admits a structure of an algebraic groupof infinite dimension with the natural filtration: G = ∪∞ d=1 G≤d where G≤d is the affinealgebraic variety of all the polynomial automorphisms of degree ≤ d of the affine spaceAnC . He also calculated its Lie algebra. So, he proved that the group Aut(AnC ) is not simpleas an abstract group. A natural and simple question asked is: Does the Cremona group Cr(n) admit astructure that is of an algebraic group of infinite dimension. This is still an open questionbecause we don’t know if the set Cr≤d (n) of birational maps of degree ≤ d admits astructure of the algebraic variety. However, the answer is yes for the set Crd (n) of allbirational maps of degree d of the projective space Pnk and PGL(n + 1)Crd (n). Theseresults are related to my PhD thesis (see [5]) that was successfully defended in 2009 at theUniversité 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 spacePnk 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)-variablesx0 , . . . , xn and are mutually prime. The common degree of Pi is called the degree ofφ; denoted deg φ. In the language of linear systems; giving a rational map such as φ isequivalent 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)| andthe 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 ; itis only the map defined in its domain of definition Dom(φ) = Pnk \V (P0 , . . . , Pn ). We saythat φ is dominant i ...