Specialization of Cremona transformations

Abstract

1. Cremona Transformations. Let V = V 2n be the Segre product variety S n x S n ' of two /^-dimensional complex projective spaces. Then any Cremona transformation T of S n into S n ' (regarded as an irreducible algebraic system of oo" ordered pairs of points) is represented on V by an irreducible ^-dimensional subvariety H which satisfies (on V) the algebraic equivalencewhere S tiJ is a subvariety S f x S/ of V and mi, ...,m n _ 1 are positive integers. We call m u ..., w n _! the characters of T, noting that, numerically,Let (<£>) and OP) be the homaloidal systems of primals in S n and S n ' associated with T; and let <&' and *P J denote the free intersection varieties of sets of / independent generic primals of (<D) and QV), proper transforms by T' 1 and T of generic spaces [n -i] of S n ' and S n respectively. Then m t = order of $ ' = order of ¥ " -' (/ = 1, ..., n -1).We envisage the primary classification of Cremona transformations of S n into S n ' as being equivalent to the identification, for each set of values of m u ...,m n^u of the distinct maximal irreducible systems of varieties H (satisfying (1.1)) on V, such that their generic members are irreducible. The number of these systems is necessarily finite. The generic member of any one of these systems will represent a transformation (defined over some extension of the complex field) which we call a primitive Cremona transformation. Plainly, therefore, the classification envisaged above may be effected by identifying the (finite number of) primitive Cremona transformations corresponding to any given set of characters m u ..., m n^1 . Every Cremona transformation with the given characters is then a specialization of one (or more) of these primitive Cremona transformations and may accordingly be said to belong to one (or more) primitive types.,f The transformations that belong to more than one primitive type could then be further classified into secondary types, and so on.If T is a primitive Cremona transformation (for given m u ..., w n _ t ), H its imagevariety on V, and (H) the maximal system defined by H, then almost all members of (H) are irreducible and represent Cremona transformations having precisely t For n -2, a Cremona transformation has only one character m-its order. There are two primitive types of transformation of order 4, generated respectively by homaloidal nets C 4 [2 3 , I 3 ] and C^P 1 , I s ]; and the associated image surfaces on V t 6 -S 2 xS 2 ' are rational surfaces-each a 3 F 10 [8]-mapped on the plane by systems C 4 [l 6 ] and C 5 P \ I 6 ] respectively. These surfaces belong to irreducible systems on V 4 6 of dimensions 20 and 22. For n = 3, a Cremona transformation has two characters m x and m 2 (the orders of its homaloidal systems in 5^ and S 3 '). For m, = m 2 = 3, Miss Hudson [1; 447] has listed four types of transformation, defining families which are easily computed to be of dimensions 39, 38, 34 and 31; and these appear in fact to be primitive types in the sense we have described. [MATHEMATIKA 15 (1968), 171-177] 4

Homaloidal determinants

“…Now, with the present data, by the associativity formula one has l( R P J P ) = e(R/J) -the multiplicity of R/J. To compute the latter we deploy the numerator of the Hilbert series of R/J in terms of the graded Betti numbers of R/J as in (30); it obtains S(t) := 1 − (n + 1)t n−1 + t 2n−2 + nt n − t 2n−1 .…”

“…To argue for embedded primes we proceed as follows. Let as above ψ denote the tail map of (30). Since R/J has homological dimension 3, any Q ∈ Ass(R/J) has codimension at most 3 and a prime Q of codimension 3 containing J is an associated prime of R/J if and only if Q ⊃ I 1 (ψ) (see, e.g., [12,Corollary 20.14(a)] for the last part).…”

Special birational transformations of projective spaces

“…48) and there are no integer solutions (d, e). If z = 14 then e < 56 an hence (d, e) ∈ {(8, 17), (12,33), (13,36), (16,39)}. If (d, e) = (12, 33) then we get (iv).…”

“…In this paper we study birational transformations Φ : P r Z with smooth irreducible and reduced base locus X ⊂ P r from the complex projective space onto a prime Fano manifold Z, thus extending the study of the classical special Cremona transformations initiated in [36], [37] and [38], and more recently and systematically revisited in [9] (see also [23] and [20]), [13] and [10]. As there are no small contractions involved, these birational transformations are among the most elementary and special links of type II between Mori fibre spaces (in the sense of the Sarkisov program), so maybe they also deserve some attention from this point of view.…”

Some Extremal Contractions Between Smooth Varieties Arising From Projective Geometry

“…The guiding principle was the beginning of Semple and Tyrrell's paper [28]: any Cremona transformation dened by a homaloidal system of primals with a single irreducible non-singular base variety is a rare enough phenomenon to merit special study. We were deeply inuenced by the classical papers and books [29,28,27,26,25,21,30,12] and the more recent [11].We conclude by remarking that even if these examples could have some interest for higher-dimensional geometry from the point of view of contractions, the methods we used are completely elementary and geometric and all the results can be considered classical or generalizations of them. Last but not least, we claim no originality for the geometrical interpretations of these classical examples.…”