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Local multiplicities of tangential trisecants to space curves

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1436)

Abstract

Let C be a curve in P 3K , where K is an algebraically closed field of characteristic zero. Assume that C is irreducible, reduced, and that C has no cusps. It is then known that C possesses finitely many tantential trisecants, and that the number of tangential trisecants is 2(d-2)(d-3)+2g(d-6) where d is the degree of C, and g is the geometric genus. The last statement is only correct if one counts tangential trisecants, including flexes, nodes, bitangents etc., with their proper multiplicities. We show how one can do this by studying an intersection product of the weak diagonal and a certain determinantal variety in the third symmetric product of the normalization of C.

If C possesses cupsps, then there will be an excess component of intersection for last product. We use the set-up from [F] to show how one can find the contribution of cusps to the global number of tangential trisicants.

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© 1990 Springer-Verlag

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Johnsen, T. (1990). Local multiplicities of tangential trisecants to space curves. In: Xambó-Descamps, S. (eds) Enumerative Geometry. Lecture Notes in Mathematics, vol 1436. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0084042

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  • DOI: https://doi.org/10.1007/BFb0084042

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-52811-1

  • Online ISBN: 978-3-540-47154-7

  • eBook Packages: Springer Book Archive