On Plane Cremona Transformations of Fixed Degree

Abstract

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