Cremona transformations of plane configurations of 6 points

Remziye Arzu Zabun

doi:10.3906/mat-1807-6

Abstract

We analyze how a set of 6 points of $\Rp 2$ in general position changes under quadratic Cremona transformations based at triples of points of the given six. As an application, we give an alternative approach to determining the deformation types (i.e. icosahedral, bipartite, tripartite and hexagonal) of 36 real Schlafli double sixes on any nonsingular real cubic surface performed by Segre.

Highlights

We call a set of six distinct points of P2 in general position a typical 6 -point configuration; ”generality” here means that no triple of points is collinear and all six are not coconic
In our recent work [5], we intended to revisit certain aspects of these topics, where we were motivated by a deformation classification of such 6-point configurations in RP2, by searching for a relation between Segre’s deformation classification of real Schläfli double sixes and Mazurovskiĭ’s deformation classification of six skew lines in RP3
The special examples of plane Cremona transformations are the quadratic Cremona transformations, Crijk, given as follows: blowing up projective plane P2 at three points pi, pj, pk and blowing down the proper transformations of three lines passing through each pair of these points

Summary

Motivation and the principal result

We call a set of six distinct points of P2 in general position a typical 6 -point configuration; ”generality” here means that no triple of points is collinear and all six are not coconic. We revisit other related results that are based on a study of the behavior of typical 6 point configurations P under internal quadratic Cremona transformations. The latter are quadratic Cremona transformations generated by triples of points of P. The deformation classification of typical 6 -point configurations in RP2 was given by Segre [7], §62, who proved that there are four classes: I, II, III and IV. He used the term primary instead of typical. We mean a path in the space of typical 6-point configurations (for more details, see Subsection 2.1)

Hexagonal QC61

Elliptic and hyperbolic lines