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A note on k-very ampleness of a bundle on a blown up plane. (English) Zbl 1079.14014
Let $$\pi : X \to \mathbb P^2$$ be the blowing-up at $$r$$ general points $$P_1,\dots ,P_r$$ and $$E_i:= \pi ^{-1}(P_i)$$ the exceptional divisors. Fix integers $$k\geq 2$$ and $$d \geq 2k+3$$. Set $$L:= \pi ^\ast (\mathcal {O}_P(d))(-E_1- \cdots -E_r))$$. Since $$L\cdot E_i = 1$$, $$L$$ is not $$k$$-ample. Take an “admissible ” $$0$$-cycle on $$X$$, i.e. assume $$\sum _{i=1}^{r} l(Z\cap E_i) \leq 2$$. Here the author gives a sharp criterion for the $$k$$-very ampleness of $$L| Z$$.
##### MSC:
 14C20 Divisors, linear systems, invertible sheaves 14J26 Rational and ruled surfaces 14N05 Projective techniques in algebraic geometry 14E25 Embeddings in algebraic geometry
##### Keywords:
$$k$$-very ample line bundle; blowing-up of the plane
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