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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. | 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 gt 2 Ivan Pan proved a result following Hudson s work on the .