Scrolls and Quartics

Abstract

In [S 1] Saint-Donat shows how to apply a theorem of Del Pezzo and Bertini (quoted as theorem l below) to recover the main result of [XXX] concerning the projective classification of codimension two cubic varieties. In this paper we show how the samc theorem, and sorne relatcd results, can be used to produce an "enumcration" of quartic varieties somcwhat more cxplicit than that given by Swinncrton-Dyer in lS2]. Our main rcsult esscntially says that a codimension 2 quartic variety which is contained in a unique quadric is rationally ruled, so that, by a theorem of Bertini, must be the projection uf a quartic scroll (see theorems 5 ami 6 bclow for complete statements).