Mathematical Subject Classification 14E07 14C17 14M25

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2000 Mathematical Subject Classification : 14E07, 14C17, 14M25 ON CHARACTERISTIC CLASSES OF DETERMINANTAL CREMONA TRANSFORMATIONS GERARD GONZALEZ-SPRINBERG AND IVAN PAN Abstract. We compute the multidegrees and the Segre numbers of general determinantal Cremona transformations, with generically re- duced base scheme, by specializing to the standard Cremona transfor- mation and computing its Segre class via mixed volumes of rational polytopes. Dedicated to the memory of Shiing-Shen Chern 1. Introduction Let F be a birational map of degree n of the projective space Pn, given by the maximal minors of a (n + 1) ? n matrix with general linear forms as entries, over an algebraically closed field K. Such a determinantal Cremona transformation may be defined geometrically by n correlations in general position and they have been considered in the classical literature on Cremona transformations by several authors, e. g. [2], [1] and [15, Chap. VIII, 4]. The family of base schemes of determinantal maps, not necessarily birational, may be identified as an open and connected subset of the Hilbert scheme of the 2-codimensional arithmetically Cohen-Macaulay subschemes of Pn (see [6]). In this article we compute the multidegrees of such F and the Segre classes of its base scheme BF , by specializing to the standard Cremona transformation Sn := (X1X2 · · ·Xn : · · · : X0 · · ·?Xi · · ·Xn : · · · : X0X1 · · ·Xn?1) and by applying methods of toric geometry.

• let


• notation let


• base scheme


• through ?e


• choose general


• general determinantal


• normal cone

Sujets

• Convex cone

Informations

• Publié par: pefav
• Nombre de lectures: 15
• Langue: English

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2000 Mathematical Subject Classification14C17, 14M25: 14E07, ON CHARACTERISTIC CLASSES OF DETERMINANTAL CREMONA TRANSFORMATIONS

GERARD GONZALEZSPRINBERG AND IVAN PAN

Abstract.We compute the multidegrees and the Segre numbers of general determinantal Cremona transformations, with generically re duced base scheme, by specializing to the standard Cremona transfor mation and computing its Segre class via mixed volumes of rational polytopes.

Dedicated to the memory of Shiing-Shen Chern

1.Introduction n LetFbe a birational map of degreenof the projective spaceP, given by the maximal minorsof a (n+ 1)×nrmfoarnerintsesa,semattighirwxlailnere over an algebraically closed fieldK. Such adeterminantal Cremona transformation may be defined geometrically bynoitisoplarenegnivehaeythndnaionselatcorr been considered in the classical literature on Cremona transformations by several authors, e. g. [2], [1] and [15, Chap. VIII,§4]. The family of base schemes of determinantal maps, not necessarily birational, may be identified as an open and connected subset of the Hilbert scheme of the n 2-codimensional arithmetically Cohen-Macaulay subschemes ofP(see [6]). In thisarticle we compute themultidegreesof suchFand theSegre classesof its base schemeB F , by specializing to thestandard Cremona transformation

 S n := (X 1 X 2 ∙ ∙ ∙X n :∙ ∙ ∙:X 0 ∙ ∙ ∙X i ∙ ∙ ∙X n :∙ ∙ ∙:X 0 X 1 ∙ ∙ ∙X n−1 )

and by applying methodsof toric geometry. In thisway it translatesinto computing mixed volumes of some special polytopes with integer vertices. The sequence of multidegrees (d 0 , . . . , d k , . . . , d n ), classically called the “type” n n    of a Cremona transformationT:P P, are given by the degreesof the n−k (direct) strict transforms byTof generalk-dimensional linear subvarietiesHof n PThe multidegrees; for a reference on this “type” via intersection theory see [16]. n are closely related (Prop. 5) to theSegre classofT,s(B T ,P), defined asthe n inverse of the Chern class of the normal bundle of the embeddingi:B T →P of the base schemeB T ofT, ifB T The general definitionsisregularly embedded. are given later. ThisSegre claslivesin the Chow groupA  (B T ) ofB T ; we also  n k consider its images(B T ) inA  (P) and theSegre numberss k =s(B T )∙H. The main results are the following:

n Theorem 1.The determinantal Cremona transformations ofPwith generically reduced base scheme and the standard Cremona transformationS n have the same multidegrees and Segre numbers. 1