Notes on embeddings of blowups
In the paper under review very ampleness and \(k\)-jet ampleness for blow-up projective complex varieties are studied. \(k\)-jet ampleness is a higher order embedding notion formerly introduced and studied by \textit{M. C. Beltrametti} and \textit{A. J. Sommese} in: ``Complex analysis and geometry'', 335-376 (1993; Zbl 0806.14014). A line bundle \(L\) on a projective variety \(X\) is \(k\)-jet ample if, given any distinct points \(x_1,\dots, x_r\) on \(X\) and integers \( k_1+ \cdots +k_r = k\), the restriction map \(\Gamma(L) \to \bigoplus_{i=1}^r\Gamma(L/\mathfrak m _{i} ^{k _i}\cdot L)\) is onto. The above notion is a natural generalization of ampleness (\(L\) is spanned if and only if it is 0-jet ample, and \(L\) is very ample if and only if it is 1-jet ample). The paper proves a rather general embedding result: Given a very ample sheaf \(\mathcal O(1)\) on a projective variety \(V\) and a closed subscheme \(X\) defined by an ideal sheaf \(\mathcal J\), if \(\mathcal J(t)\) is spanned for some positive integer \(t\), then \(\pi^\ast \mathcal J(s)\) is very ample on the blowing-up \(\pi\colon Z\to V\) along \(X\) for every integer \(s\geq t\). The theorem provides very precise results in some particular cases: minimal degree projective varieties, complete intersections or determinantal varieties. A generalization of this theorem for \(k\)-jet ampleness is also given: If in addition to the former hypotheses one assumes that \(\mathcal O(1)\) is \(\omega\)-jet ample, then \(\pi^\ast\mathcal O(a)-\mathcal O(-b\pi^{-1}(X))\) is \(k\)-jet ample for \(a\geq bt+k/\omega\) and \(b\geq k\). Nice applications to the case when \(L=\mathcal O(k)\) and one blows up a disjoint union of subvarieties of the projective space and also to the case of Hirzebruch and del Pezzo surfaces are given.
scientific article; zbMATH DE number 1013957