Examples of special quadratic birational transformations into complete intersections of quadrics

TL;DR

This paper constructs explicit examples of quadratic birational transformations into complete intersections of quadrics, addressing open existence and factoriality questions using computer algebra methods.

Contribution

It provides the first explicit constructions of certain quadratic birational transformations with factorial images, solving previously open existence problems.

Findings

Constructed four explicit examples of transformations

Confirmed factoriality of images in two cases

Demonstrated existence of previously unknown transformations

Abstract

In our previous works (2012, 2013), we provided a finite list of properties characterizing all potential types of quadratic birational transformations of a projective space into a factorial variety, whose base locus is smooth and irreducible. However, some existence problems remained open. Among them one had to prove that the image of a given transformation was factorial, but in two particular situations, even the mere existence of the transformation was left as an open problem. In this paper, we use computer algebra methods to construct explicitly four examples of such transformations; two of them were among those for which it was not known that the image was factorial, and the other two show the existence of the two above referred transformations.