Higher order embeddings of certain blow-ups of $\mathbb {P}^2$
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- by Cindy De Volder and Halszka Tutaj-Gasińska
- Proc. Amer. Math. Soc. 137 (2009), 4089-4097
- DOI: https://doi.org/10.1090/S0002-9939-09-10037-0
- Published electronically: July 10, 2009
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Abstract:
Let $X_n$ be the blow-up of the projective plane along $n$ general points of a smooth cubic plane curve and let $\mathcal {L}$ be the linear series of strict transforms of plane curves of degree $d$ having multiplicity at least $m_i$ at the $i$-th blown-up point. We prove that if $\mathcal {L}$ is $k$-very ample, then $\mathcal {L}$ is excellent and $\mathcal {L}\cdot (-K_n) \geq k+2$. Then we give a numerical criterion for the $k$-very ampleness of excellent classes with $\mathcal {L} \cdot (-K_n) \geq k+2$, which in many cases is a necessary and sufficient condition.References
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Bibliographic Information
- Cindy De Volder
- Affiliation: Department of Pure Mathematics and Computer Algebra, Ghent University, Krijgslaan 281, S22, B-9000 Ghent, Belgium
- Email: cindy.devolder@ugent.be
- Halszka Tutaj-Gasińska
- Affiliation: Institute of Mathematics, Jagiellonian University, Łojasiewicza 6, PL-30348 Kraków, Poland – and – Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, PL-00956 Warszawa, Poland
- MR Author ID: 612578
- Email: htutaj@im.uj.edu.pl
- Received by editor(s): May 17, 2008
- Received by editor(s) in revised form: January 24, 2009, and April 30, 2009
- Published electronically: July 10, 2009
- Communicated by: Ted Chinburg
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 137 (2009), 4089-4097
- MSC (2000): Primary 14C20
- DOI: https://doi.org/10.1090/S0002-9939-09-10037-0
- MathSciNet review: 2538570