On quasi-complete intersections of codimension 2

Choi, Youngook

doi:10.1090/S0002-9939-05-08425-X

On quasi-complete intersections of codimension $2$
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by Youngook Choi

Proc. Amer. Math. Soc. 134 (2006), 1249-1256

DOI: https://doi.org/10.1090/S0002-9939-05-08425-X

Published electronically: December 14, 2005

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Abstract:

In this paper, we prove that if $X\subset \mathbb {P}^n$, $n\ge 4$, is a locally complete intersection of pure codimension $2$ and defined scheme-theoretically by three hypersurfaces of degrees $d_1\ge d_2\ge d_3$, then $H^1(\mathbb {P}^n,\mathcal {I}_X(j))=0$ for $j<d_3$ using liaison theory and the Arapura vanishing theorem for singular varieties. As a corollary, a smooth threefold $X\subset \mathbb {P}^5$ is projectively normal if $X$ is defined by three quintic hypersurfaces.

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