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Higher order embeddings of certain blow-ups of $ \mathbb{P}^2$

Authors: Cindy De Volder and Halszka Tutaj-Gasinska
Journal: Proc. Amer. Math. Soc. 137 (2009), 4089-4097
MSC (2000): Primary 14C20
Published electronically: July 10, 2009
MathSciNet review: 2538570
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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.

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Additional Information

Cindy De Volder
Affiliation: Department of Pure Mathematics and Computer Algebra, Ghent University, Krijgslaan 281, S22, B-9000 Ghent, Belgium

Halszka Tutaj-Gasinska
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

Keywords: $k$-very ample, anticanonical rational surface
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
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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