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- W1823641945 abstract "Let $X^nsubset C^{n+a}$ or $X^nsubset P^{n+a}$ be a patch of an analytic submanifold of an affine or projective space, let $xin X$ 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 $Lsubset X$ in the following cases: k=1 and m=n+1; k=n-1, $ageq 2$, and m=2; $ngeq 4$, 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." @default.
- W1823641945 created "2016-06-24" @default.
- W1823641945 creator A5068623131 @default.
- W1823641945 date "1996-12-24" @default.
- W1823641945 modified "2023-09-27" @default.
- W1823641945 title "Is a linear space contained in a variety? - On the number of derivatives needed to tell" @default.
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