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Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie

IMPACT FACTOR 2018: 1.859

CiteScore 2018: 1.14

SCImago Journal Rank (SJR) 2018: 2.554
Source Normalized Impact per Paper (SNIP) 2018: 1.411

Mathematical Citation Quotient (MCQ) 2017: 1.49

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Volume 1999, Issue 508


Is a linear space contained in a submanifold? – On the number of derivatives needed to tell

J. M Landsberg
Published Online: 2008-06-11 | DOI: https://doi.org/10.1515/crll.1999.508.53


Let Xn or Xn ⊂ ℙ n + a be a patch of a C submanifold of an affine or projective space such that through each point xX there exists a line osculating to order n + 1 at x. We show that X is uniruled by lines, generalizing a classical theorem for surfaces. We describe two circumstances that imply linear spaces of dimension k osculating to order two must be contained in X, shedding light on some of Ein's results on dual varieties. We present some partial results on the general problem of finding the integer m 0 = m 0(k, n, a) such that there exist examples of patches Xn ⊂ ℙn + a, having a linear space L of dimension k osculating to order m 0 — 1 at each point such that L is not locally contained in X, but if there are k-planes osculating to order m 0 at each point, they are locally contained in X.

The same conclusions hold in the analytic category and complex analytic category if there is a linear space osculating to order m at one general point xX.

About the article

Received: 1998-05-15

Published Online: 2008-06-11

Published in Print: 1999-03-12

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 1999, Issue 508, Pages 53–60, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: https://doi.org/10.1515/crll.1999.508.53.

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