Pencils on an algebraic variety and a new proof of a theorem of Bertini
Oscar Zariski · Transactions of the American Mathematical Society · 1941
OSCAR ZARISKI(') Introduction.A well known theorem of Bertini-Enriques on reducible linear systems of Fr_i's on an algebraic Vr (i.e., linear systems in which each element is a reducible VT-i) 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-i 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