Cremona transformations associated to quadratic complexes
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Cremona transformations associated to quadratic complexes D. Avritzer · H. Lange
Received: 16 April 2012 / Accepted: 27 July 2012 / Published online: 15 February 2013 © Springer-Verlag Italia 2013
Abstract To a general quadratic line complex and any pair of different lines in it one can associate a cubo-cubic Cremona transformation of projective 3-space in a natural way. This gives a map γ from the space of pairs of lines in general quadratic complexes to the space of equivalence classes of these cubo-cubic transformations. We compute the dimension of a general fibre of γ as well as the codimension of subspace of the transformations arizing in this way in the corresponding space of Cremona transformations. Keywords
Cremona transformations · Quadratic complex · Line complex
Mathematics Subject Classification
14E67
1 Introduction In the paper [2] the authors associate to every triple (X, L 1 , L 2 ), where X is a general quadratic line complex and L 1 and L 2 are different lines in X , a cubo-cubic Cremona transformation ϕ = ϕ(X,L 1 ,L 2 ) : P3 → P3 . It is determinantal if L 1 ∩ L 2 = ∅ and de Jonquières otherwise. For the definitions see Sects. 2 and 3 below. We call two Cremona transformations P3 → P3 equivalent if they differ at most by an automorphism of the image-P3 . The equivalence classes of determinantal (respectively de Jonquières) cubo-cubic Cremona transformations are parametrized by an irreducible variety Cr det (3, 3) (respectively Cr d J (3, 3)) of dimension
H. Lange would like to thank DAAD (Germany) and FAPEMIG (Brasil) for support during the preparation of this paper. D. Avritzer Departamento de Matemática, UFMG, Belo Horizonte, MG 30161-970, Brazil e-mail: [email protected] H. Lange (B) Department Mathematik, Universität Erlangen-Nürnberg, Nürnberg, Germany e-mail: [email protected]
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24 (respectively 23) (see [6]). On the other hand, it is easy to see that the triples (X, L 1 , L 2 ) with L 1 ∩ L 2 = ∅ (respectively L 1 ∩ L 2 = point) are parametrized by an irreducible variety Bdet of dimension 23 (respectively C d J of dimension 22). So we get maps γ = γ det : Bdet → Cr det (3, 3)
γ = γ d J : C d J → Cr d J (3, 3).
and
A natural question seems to be: What is the codimension of the image of γ ? Or equivalently: What is the dimension of a general fibre of γ ? The following theorem is the main result of this note. Theorem 1. In both cases a general fibre of γ is of dimension 1. 2. The image of γ is of codimension 2 in Cr det (3, 3) respectively Cr d J (3, 3). The idea of the proof is as follows: The base locus of a determinantal or de Jonquières transformation has Hilbert polynomial p(n) = 6n−2. Taking the base locus gives morphisms Bs : Cr det (3, 3) → Hilb p (P3 )
and
Bs : Cr d J (3, 3) → Hilb p (P3 ).
In both cases we consider the composed morphism Bs ◦γ . The main point in the proof is to show that Bs is generically injective on Im(γ ). Since one knows the image of the composed map, it suffices to work out the dimension of this image and for this
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