Invariants of pencils of binary cubics

Abstract

Let U denote the variety of pencils of binary cubics without base point defined over an algebraically closed field k whose characteristic is not equal to 2 or 3. (For full details of the terminology, see § 2.) There is a natural action of SL(2) on U; our main object is to show that U possesses a good quotient for this action in the sense of ((7), Definition 1·5) or ((4), p. 70), and to identify this quotient with the affine line over k. In fact, we proveTheorem 1. There exists a morphism φ: such that (, φ) is a good quotient of U by SL(2). Moreover, all but one of the fibres of φ are orbits; the exceptional fibre consists of two orbits corresponding to pencils with one or two triple points.