Higher order embeddings of certain blow-ups of ℙ²

De Volder, Cindy; Tutaj-Gasińska, Halszka

doi:10.1090/S0002-9939-09-10037-0

Higher order embeddings of certain blow-ups of $\mathbb {P}^2$
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by Cindy De Volder and Halszka Tutaj-Gasińska PDF

Proc. Amer. Math. Soc. 137 (2009), 4089-4097 Request permission

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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