Very Ample Linear Systems on Blowings-Up at General Points of Projective Spaces

Coppens, Marc

Canadian mathematical bulletin, Tome 45 (2002) no. 3, pp. 349-354

Résumé

Let ${{\mathbf{P}}^{n}}$ be the $n$ -dimensional projective space over some algebraically closed field $k$ of characteristic 0. For an integer $t\,\ge \,3$ consider the invertible sheaf $O\left( t \right)$ on ${{\mathbf{P}}^{n}}$ (Serre twist of the structure sheaf). Let $N\,=\,\left( \underset{n}{\mathop{t+n}}\, \right)$ , the dimension of the space of global sections of $O\left( t \right)$ , and let $k$ be an integer satisfying $0\,\le \,k\,\le \,N\,-\,\left( 2n\,+\,2 \right)$ . Let ${{P}_{1}},\ldots ,{{P}_{k}}$ be general points on ${{\mathbf{P}}^{n}}$ and let $\pi :\,X\,\to \,{{\mathbf{P}}^{n}}$ be the blowing-up of ${{\mathbf{P}}^{n}}$ at those points. Let ${{E}_{i}}\,=\,{{\pi }^{-1}}\left( {{P}_{i}} \right)$ with $1\,\le \,i\,\le \,k$ be the exceptional divisor. Then $M={{\pi }^{*}}\left( O(t) \right)\,\otimes \,{{O}_{X}}\left( -{{E}_{1}}-\cdots -{{E}_{k}} \right)$ is a very ample invertible sheaf on $X$ .

Détail

DOI : 10.4153/CMB-2002-037-0

Mots-clés : 14E25, 14N05, 14N15, blowing-up, projective space, very ample linear system, embeddings, Veronese map