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Local properties of secant varieties in symmetric products. I


Authors: Mark E. Huibregtse and Trygve Johnsen
Journal: Trans. Amer. Math. Soc. 313 (1989), 187-204
MSC: Primary 14H45; Secondary 14B12, 14M15, 14N10
DOI: https://doi.org/10.1090/S0002-9947-1989-0929672-X
MathSciNet review: 929672
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ L$ be a line bundle on an abstract nonsingular curve $ C$, let $ V \subset {H^0}(C,L)$ be a linear system, and denote by $ {C^{(d)}}$ the symmetric product of $ d$ copies of $ C$. There exists a canonically defined $ {C^{(d)}}$-bundle map:

$\displaystyle \sigma :V \otimes {\mathcal{O}_{{C^{(d)}}}} \to {E_L},$

where $ {E_L}$ is a bundle of rank $ d$ obtained from $ L$ by a so-called symmetrization process. The various degenerary loci of $ \sigma $ can be considered as subsecant schemes of $ {C^{(d)}}$. Our main result, Theorem 4.2, is given in $ \S4$, where we obtain a local matrix description of $ \sigma $ valid (also) at points on the diagonal in $ {C^{(d)}}$, and thereby we can determine the completions of the local rings of the secant schemes at arbitrary points. In $ \S5$ we handle the special case of giving a local scheme structure to the zero set of $ \sigma $.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1989-0929672-X
Keywords: Secant varieties of curves, local geometry
Article copyright: © Copyright 1989 American Mathematical Society

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