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Generic inner projections of projective varieties and an application to the positivity of double point divisors


Author: Atsushi Noma
Journal: Trans. Amer. Math. Soc. 366 (2014), 4603-4623
MSC (2010): Primary 14N15, 14N05
DOI: https://doi.org/10.1090/S0002-9947-2014-06129-1
Published electronically: May 12, 2014
MathSciNet review: 3217694
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Abstract: Let $ X\subseteq \mathbb{P}^{N}$ be a smooth nondegenerate projective variety of dimension $ n\geq 2$, codimension $ e$ and degree $ d$ with the canonical line bundle $ \omega _{X}$ defined over an algebraically closed field of characteristic zero. The purpose here is to prove that the base locus of $ \vert\mathcal {O}_{X}(d-n-e-1)\otimes \omega _{X}^{\vee }\vert$ is at most a finite set, except in a few cases. To describe the exceptional cases, we classify (not necessarily smooth) projective varieties whose generic inner projections have exceptional divisors. As applications, we prove the $ (d-e)$-regularity of $ \mathcal {O}_{X}$, Property $ (N_{k-d+e})$ for $ \mathcal {O}_{X}(k)$, and inequalities for the delta and sectional genera.


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Additional Information

Atsushi Noma
Affiliation: Department of Mathematics, Yokohama National University, Yokohama 240-8501 Japan
Email: noma@ynu.ac.jp

DOI: https://doi.org/10.1090/S0002-9947-2014-06129-1
Keywords: Linear projection, inner projection, double point divisor, ramification divisor
Received by editor(s): September 28, 2011
Received by editor(s) in revised form: August 23, 2012
Published electronically: May 12, 2014
Additional Notes: The author was partially supported by Grant-in-Aid for Scientific Research, Japan Society for the Promotion of Science
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.