Is a linear space contained in a variety? - On the number of derivatives needed to tell

Abstract

Let $X^n\subset C^{n+a}$ or $X^n\subset P^{n+a}$ be a patch of an analytic submanifold of an affine or projective space, let $x\in X$ be a general point, and let L^k be a linear space of dimension k osculating to order m at x. If m is large enough, one expects L to be contained in X and thus X contains a linear space of dimension kthrough almost every point. We show that $L\subset X$ in the following cases: k=1 and m=n+1; k=n-1, $a\geq 2$, and m=2; $n\geq 4$, k=n-2 and m=4. We prove these results by first deriving the order of osculation that generically implies containment and then showing that in these cases containment must occur. If X is a patch of a projective variety, we address the question as to whether X can be a smooth variety. We show that if there is a P^k through each point and $codim(X)<\frac{k}{n-k}$ then X cannot be a smooth variety.