Abstract
For a nondegenerate curve X in projective space PN and 1≤n≤N define the n-secant variety Secn(X) as the closure of the union of all (n-1)-dimensional linear spaces in PN containing n smooth points of X. It is proven that Secn(X) alway has the expected dimension. A consequence is Nagata's theorem on ruled surfaces.
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Lange, H. Higher secant varieties of curves and the theorem of Nagata on ruled surfaces. Manuscripta Math 47, 263–269 (1984). https://doi.org/10.1007/BF01174597
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DOI: https://doi.org/10.1007/BF01174597