Theorems of Bertini on Linear Systems

Abstract

As the fundamental $theorem_{\overline{3}}$ of the classical algebraic geometry we have these of Bertini: I. $T/le$ general section $U_{-1}$ of an algebraic variety $U_{r}$ by a linear system $with\ell ut$ fixed components is irreducible, provided $t\prime_{l}at$ tfte linear system is not composed of an algebraic $pe\prime\prime cil$ . II. Tlte general section $U_{r-1}$ of $U_{r}$ by a linear system can not lave any singular points outside the singlar points of $U_{r}$ and oulside the base points of the linear system.