Pencils on an algebraic variety and a new proof of a theorem of Bertini

Introduction. A well known theorem of Bertini-Enriques on reducible linear systems of Vr_-'s on an algebraic Vr (i.e., linear systems in which each element is a reducible Vri-) states that any such system, if free from fixed components, is composite with a pencil. The usual geometric proof of this theorem is based on another theorem of Bertini, to the effect that the general Vr_1 of a linear system cannot have multiple points outside the singular locus of the variety Vr and the base locus of the system. This geometric proof has been subsequently completed and presented by van der Waerden under an algebraic form [3]. In this paper we give a new proof of the theorem of Bertini on reducible linear systems and we also extend this theorem to irrational pencils, i.e., pencils of genus p >0. Our proof does not make use of the second theorem of Bertini just quoted. In the case of pencils (linear-or irrational), we first observe that a pencil { WI on Vr is defined by a field P of algebraic functions of one variable which is a subfield of the field z of rational functions on Vr. The whole proof is then essentially based on the simple remark that the pencil I W} is composite with another pencil { W }, defined by a field P, if and only if P is a subfield of P. This property is a straightforward consequence of the geometric definition of composite pencils. As a matter of fact we prefer to define composite pencils by this property. At any rate, it is then true that a pencil I W} is non-composite if and only if the corresponding field P is maximally algebraic in Z. In the light of this approach to the question, the theorem of Bertini on reducible pencils is almost a direct consequence of the well known fact that an irreducible algebraic variety V, over a ground field K, is absolutely irreducible if K is maximally algebraic in the field of rational functions on V. In the case of linear systems of dimension > 1 the proof is even simpler, provided use is made of a certain lemma (Lemma 5). This lemma is, however, of interest in itself. A sizable portion of the paper (Part I) is devoted to the development of the concept of a pencil and of the basic properties of pencils in the abstract case of an arbitrary ground field (of characteristic zero).