On plane Cremona transformations of fixed degree

TL;DR

This paper investigates the structure and connectivity of the variety of plane Cremona transformations of fixed degree, providing dimension calculations, irreducible component decompositions, and connectivity results.

Contribution

It offers a detailed analysis of the geometry and topology of the space of plane Cremona transformations, including new results on their connectedness and component structure.

Findings

Bir_d is connected for all degrees d.

Bir_d^o is connected for degrees d less than 7.

Dimensions and irreducible components of Bir_d and Bir_d^o are explicitly computed.

Abstract

We study the quasi-projective variety Bir_d of plane Cremona transformations defined by three polynomials of fixed degree d and its subvariety Bir_d^o where the three polynomials have no common factor. We compute their dimension and the decomposition in irreducible components. We prove that Bir_d is connected for each d and Bir_d^o is connected when d < 7.