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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Local properties of secant varieties in symmetric products. II


Author: Trygve Johnsen
Journal: Trans. Amer. Math. Soc. 313 (1989), 205-220
MSC: Primary 14H45; Secondary 14B12, 14M15, 14N10
MathSciNet review: 929673
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Abstract: Let $ V$ be a linear system on a curve $ C$. In Part I we described a method for studying the secant varieties $ V_d^r$ locally. The varieties $ V_d^r$ are contained in the $ d$-fold symmetric product $ {C^{(d)}}$.

In this paper (Part II) we apply the methods from Part I. We give a formula for local tangent space dimensions of the varieties $ V_d^1$ valid in all characteristics (Theorem 2.4).

Assume $ \operatorname{rk}\;V = n + 1$ and $ \operatorname{char} K = 0$. In $ \S\S3$ and $ 4$ we describe in detail the projectivized tangent cones of the varieties $ V_n^1$ for a large class of points. The description is a generalization of earlier work on trisecants for a space curve.

In $ \S5$ we study the curve in $ {C^{(2)}}$ consisting of divisors $ D$ such that $ 2D \in V_4^1$ . We give multiplicity formulas for all points on this curve in $ {C^{(2)}}$ in terms of local geometrical invariants of $ C$. We assume $ \operatorname{char} K = 0$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1989-0929673-1
PII: S 0002-9947(1989)0929673-1
Keywords: Secant varieties of curves, local geometry
Article copyright: © Copyright 1989 American Mathematical Society