On Reider’s method and higher order embeddings.

*(English)*Zbl 0702.14010In this paper various higher order generalizations of very ampleness are considered. The primary generalization, k-spannedness, is defined as follows. A line bundle L on a smooth surfaces S is k-spanned for a non- negative integer k if given any length \((k+1)\), 0-dimensional subscheme (Z,\({\mathcal O}_ Z)\) of S which is curvilinear (i.e. which lies on a germ of a smooth curve), we have a surjection, \(\Gamma\) (L)\(\to \Gamma (L\otimes {\mathcal O}_ Z)\to 0\). A line bundle, L, is spanned at all points by global sections (respectively very ample) if and only if L is 0-spanned (respectively 1-very ample). A Reider type criterion is given for the adjoint bundle \(K_ S\otimes L\) to be k-spanned. This definition has the virtue that it is weaker than other reasonable definitions, and thus results proven using this definition will hold true for stronger k- th order notions of very ampleness. This has turned out to be important since a strengthened form of a conjecture stated at the end of the paper has been proved independently by different methods by the reviewers [“Zero cycles and k-th order embeddings of smooth projective surfaces”, in “Projective Surfaces and their Classification”, 1988 Cortona Proc.: Symp. Math. INDAM] and also by Reider. This result, which subsumes the Reider type criterion of the paper under review, is stated as follows.

Theorem. Let L be a nef line bundle on a smooth surface S. Let \(L\cdot L\geq 4k+1\). Then given any 0-dimensional subscheme (Z,\({\mathcal O}_ Z)\) of S of length \(\leq k\), either \(\Gamma (K_ S\otimes L)\to \Gamma (K_ S\otimes L\otimes {\mathcal O}_ Z)\to 0\) or there exists an effective divisor D such that L-2D is \({\mathbb{Q}}\)-effective, D contains some 0- dimensional subscheme \((Z',{\mathcal O}_{Z'})\) of (Z,\({\mathcal O}_ Z)\) of length \(k'\leq k\), and \(L\cdot D-k'\leq D\cdot D<L\cdot D/2<k'.\)

Because of this result, it is better to work with k-very ampleness, a more natural generalization of very ampleness, where the curvilinear \(length\)-(k\(+1)\) 0-dimensional subschemes in the definition of k- spannedness are replaced by arbitrary \(length\)-(k\(+1)\) 0-dimensional subschemes of S. The results in the paper under review on the k- spannedness of powers of the canonical bundle, hold with k-very ample used in place of k-spanned. The only change in the proofs is to use the above criterion instead of the criterion in the paper.

Some further papers on k-spannedness are (in chronological order) M. Beltrametti and A. Sommese [“On k-spannedness for projective surfaces”, Algebraic geometry, Proc. Int. Conf., L’Aquila/Italy 1988, Lect. Notes Math. 417, 24-51 (1990) and “On the relative adjunction mapping”, Math. Scand. 65, No.2, 189-205 (1989)]; E. Ballico [Proc. Am. Math. Soc. 105, No.3, 531-534 (1989; Zbl 0671.14019) and “On k-spanned projective surfaces”, in Algebraic geometry, Proc. Int. Conf., L’Aquila/Italy 1998, Lect. Notes Math. 1417, 23 (1990)]; E. Ballico and M. Beltrametti [Manuscr. Math. 61, No.4, 447-458 (1988; Zbl 0661.14005)]; F. Catanese and L. Göttsche [“d-very-ample line bundles and embeddings of Hilbert schemes of 0-cycles”, Manuscr. Math. 68, No.3, 337-341 (1990)]; M. Andreatta and M. Palleschi, “On the 2-very ampleness of the adjoint bundle” (preprint).

Theorem. Let L be a nef line bundle on a smooth surface S. Let \(L\cdot L\geq 4k+1\). Then given any 0-dimensional subscheme (Z,\({\mathcal O}_ Z)\) of S of length \(\leq k\), either \(\Gamma (K_ S\otimes L)\to \Gamma (K_ S\otimes L\otimes {\mathcal O}_ Z)\to 0\) or there exists an effective divisor D such that L-2D is \({\mathbb{Q}}\)-effective, D contains some 0- dimensional subscheme \((Z',{\mathcal O}_{Z'})\) of (Z,\({\mathcal O}_ Z)\) of length \(k'\leq k\), and \(L\cdot D-k'\leq D\cdot D<L\cdot D/2<k'.\)

Because of this result, it is better to work with k-very ampleness, a more natural generalization of very ampleness, where the curvilinear \(length\)-(k\(+1)\) 0-dimensional subschemes in the definition of k- spannedness are replaced by arbitrary \(length\)-(k\(+1)\) 0-dimensional subschemes of S. The results in the paper under review on the k- spannedness of powers of the canonical bundle, hold with k-very ample used in place of k-spanned. The only change in the proofs is to use the above criterion instead of the criterion in the paper.

Some further papers on k-spannedness are (in chronological order) M. Beltrametti and A. Sommese [“On k-spannedness for projective surfaces”, Algebraic geometry, Proc. Int. Conf., L’Aquila/Italy 1988, Lect. Notes Math. 417, 24-51 (1990) and “On the relative adjunction mapping”, Math. Scand. 65, No.2, 189-205 (1989)]; E. Ballico [Proc. Am. Math. Soc. 105, No.3, 531-534 (1989; Zbl 0671.14019) and “On k-spanned projective surfaces”, in Algebraic geometry, Proc. Int. Conf., L’Aquila/Italy 1998, Lect. Notes Math. 1417, 23 (1990)]; E. Ballico and M. Beltrametti [Manuscr. Math. 61, No.4, 447-458 (1988; Zbl 0661.14005)]; F. Catanese and L. Göttsche [“d-very-ample line bundles and embeddings of Hilbert schemes of 0-cycles”, Manuscr. Math. 68, No.3, 337-341 (1990)]; M. Andreatta and M. Palleschi, “On the 2-very ampleness of the adjoint bundle” (preprint).

Reviewer: M.Beltrametti; A.J.Sommese

##### MSC:

14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |

14C20 | Divisors, linear systems, invertible sheaves |

14E25 | Embeddings in algebraic geometry |

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\textit{M. Beltrametti} et al., Duke Math. J. 58, No. 2, 425--439 (1989; Zbl 0702.14010)

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