d-very-ample line bundles and embeddings of Hilbert schemes of 0-cycles
Let X be a complete algebraic variety over an algebraically closed field. Let \(X^{[k+1]}\) be the Hilbert scheme of 0-cycles on X of length \(k+1\). Let L be a line bundle on X. L is said to be k-very ample if the restriction map \(H^ 0(X,L)\to H^ 0(Z,{\mathcal O}_ Z(L))\) is onto for every O-cycle \(Z\in X^{[k+1]}\). E.g. O-very ample means spanned by global sections, 1-very ample means very ample. In the case of smooth surfaces this notion was recently introduced by Sommese and the reviewer. Recently, the k-very ampleness has been extensively studied by several authors. The main result of the paper is to show that if L is k-very ample, then L is \((k+1)\)-very ample if and only if the morphism \(X^{[k+1]}\to Grass(k+1,H^ 0(X,L)^*)\) is an embedding. This improves a previous result by Sommese and the reviewer.
scientific article; zbMATH DE number 4204526