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Invariants of pencils of binary cubics

1981; Cambridge University Press; Volume: 89; Issue: 2 Linguagem: Inglês

10.1017/s0305004100058102

1469-8064

P. E. Newstead,

Advanced Algebra and Geometry

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 prove Theorem 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 .