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Reducible Hilbert scheme of smooth curves with positive Brill-Noether number


Author: Changho Keem
Journal: Proc. Amer. Math. Soc. 122 (1994), 349-354
MSC: Primary 14H10; Secondary 14C05
MathSciNet review: 1221726
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Abstract: In this paper we demonstrate various reducible examples of the scheme $ \mathcal{I}{' _{d,g,r}}$ of smooth curves of degee d and genus g in $ {\mathbb{P}^r}$ with positive Brill-Noether number. An example of a reducible $ \mathcal{I}{' _{d,g,r}}$ with positive $ \rho (d,g,r)$, namely, the example $ \mathcal{I}{' _{2g - 8,g,g - 8}},$, has been known to some people and seems to have first appeared in the literature in Eisenbud and Harris, Irreducibility of some families of linear series with Brill-Noether number $ -1$, Ann. Sci. École Norm. Sup. (4) 22 (1989), 33-53. The purpose of this paper is to add a wider class of examples to the list of such reducible examples by using general k-gonal curves. We also show that $ \mathcal{I}{' _{d,g,r}}$ is irreducible for the range of $ d \geq 2g - 7$ and $ g - d + r \leq 0$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1994-1221726-3
Keywords: Hubert scheme, linear series, gonality
Article copyright: © Copyright 1994 American Mathematical Society