On inflection points , monomial curves , and hypersurfaces containing projective curves *

Let X c It w be a projective curve. In this paper we try to find an upper bound for the number of linearly independent hypersurfaces Z D X of given degree m and to investigate the borderline cases. Maybe the first result in this direction is due to Castelnuovo. To wit, his wellknown lemma (see, for instance, [8, Ch. 4, Sect. 3]) says that if n ( n 1)/2 linearly independent quadrics pass through d _> 2n + 3 points in uniform position in IP n, then these points lie on a rational normal curve. As an immediate consequence, one can see that