On Characteristic Classes of Determinantal Cremona Transformations

Abstract

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

Gluing determinantal Cremona maps

“…Even if it might seem a bit artificial (since the results about the standard Cremona maps are well known, in particular its projective degrees, and computable by other means, see [GSP06]), we illustrate an application of Proposition 1 in the example of the standard Cremona maps…”

“…Our guiding remark is that the so-called standard Cremona map (see [GSP06] for more references about the standard Cremona maps)…”

A connection between birational automorphisms of the plane and linear systems of curves

“…For higher dimension there has also been a lot of research on the subject [21][22][23][24][25][26][27][28], though the results obtained remain sporadic and, in general, there are no substantial advances with respect to the pioneering works in the knowledge either about the structure of arbitrary Cremona transformations themselves or about the structure of the group of Cremona transformations, even for n = 3. The results presented here attempt to open several ways that can be used to provide significant results concerning Cremona transformations for n ≥ 3 (see Section 4).…”

The group of Cremona transformations generated by linear maps and the standard involution

“…For n = 3, this map is called a tetrahedral transformation and written T tet in [14, p. 301, §14] or the standard cubo-cubic transformation of space [29, p. 179] or a (3,3)-transformation [1, p. 2071-2072, 2108], as its degree and the degree of its inverse are three (but of course this map is not the only one having these properties). Nowadays, the usual terminology is to call σ n , in any dimension n, the standard Cremona transformation (see for instance [12], [13], [21, p. 72], [8], [6]). The map σ n restricts to an automorphism of the standard torus T ⊂ P n k and contracts the n + 1 coordinate hyperplanes, i.e.…”