Special cubic Cremona transformations of ${\mathbb{P}}^6$ and ${\mathbb{P}}^7$

Abstract

A famous result of B. Crauder and S. Katz (1989) concerns the classification of special Cremona transformations whose base locus has dimension at most two. Furthermore, they also proved that a special Cremona transformation with base locus of dimension three has to be one of the following: 1) a quinto-quintic transformation of ${\mathbb{P}}^5$; 2) a cubo-quintic transformation of ${\mathbb{P}}^6$; or 3) a quadro-quintic transformation of ${\mathbb{P}}^8$. Special Cremona transformations as in case 1) have been classified by L. Ein and N. Shepherd-Barron (1989), while in our previous work (2013), we classified special quadro-quintic Cremona transformations of ${\mathbb{P}}^8$. The main aim here is to consider the problem of classifying special cubo-quintic Cremona transformations of ${\mathbb{P}}^6$, concluding the classification of special Cremona transformations whose base locus has dimension three.

Added an explicit example that makes the classification effective. Accepted for publication in Adv. Geom

References in corpus (4)

• Varieties with quadratic entry locus, I
• On special quadratic birational transformations of a projective space into a hypersurface
• On special quadratic birational transformations whose base locus has dimension at most three
• Examples of special quadratic birational transformations into complete intersections of quadrics

Cited by in corpus (1)

• A Macaulay2 package for computations with rational maps