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Very Ample Linear Systems on BlowingsUp at General Points of Projective Spaces


Published:20020901
Printed: Sep 2002
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Abstract
Let $\mathbf{P}^n$ be the $n$dimensional projective space over some
algebraically closed field $k$ of characteristic $0$. For an integer
$t\geq 3$ consider the invertible sheaf $O(t)$ on $\mathbf{P}^n$ (Serre
twist of the structure sheaf). Let $N = \binom{t+n}{n}$, the
dimension of the space of global sections of $O(t)$, and let $k$ be an
integer satisfying $0\leq k\leq N  (2n+2)$. Let $P_1,\dots,P_k$
be general points on $\mathbf{P}^n$ and let $\pi \colon X \to
\mathbf{P}^n$ be the blowingup of $\mathbf{P}^n$ at those points.
Let $E_i = \pi^{1} (P_i)$ with $1\leq i\leq k$ be the exceptional
divisor. Then $M = \pi^* \bigl( O(t) \bigr) \otimes O_X (E_1 
\cdots E_k)$ is a very ample invertible sheaf on $X$.