Extension of morphisms defined on a divisor
type: Item
instance of: scholarly article
title: Extension of morphisms defined on a divisor
zbMATH Open document ID: 0595.14005
DOI: 10.1007/BF01458322
author: Fernando Serrano
published in: Mathematische Annalen
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164256
review text: Let X be a smooth projective variety of dimension 2 or 3, D an effective smooth divisor on X, and \(\phi: D\to B\) a morphism onto a curve B. In this paper the author gives some sufficient conditions for the map \(\phi\) to be the restriction of a morphism \(\psi: X\to B.\) Let s be the degree of some (possibly singular) plane curve birationally isomorphic to B. With the above hypothesis the following is proved: Theorem 1: Suppose that dim X\(=2\), and \(\phi\) is a finite morphism of degree r. Set \(d=r(s-1)\). If \(D^ 2>(d+1)^ 2\) then there exists a morphism \(\psi: X\to B\) whose restriction to D is \(\phi\). Several examples show that \(theorem\quad 1\) fails in general if \(D^ 2\leq (d+1)^ 2\), but partial results are proved in the range \(4d+1\leq D^ 2\leq (d+1)^ 2\) with the further condition \(H^ 1(X,{\mathcal O}_ X)=0.\) Theorem 2: Assume that dim X\(=3\), D is a very ample divisor on X, and the fibres of \(\phi\) are connected. Denote by g the arithmetic genus of a fibre F of \(\phi\), and by \(K_ X\) the canonical divisor on X. Set \(r=D\cdot F\), \(d=r(s-1)\). If \(D^ 3>(d+1)^ 2\) and dim \(H^ 0(X,{\mathcal O}_ X(D))\geq 3r+3+2g\) then \(\phi\) is the restriction of a morphism \(\psi: X\to B.\) The proof of theorem 1 is based on a vanishing theorem for cohomology due to Miyaoka, and a generalization of Bombieri's method on numerical connectedness of divisors. Theorem 2 relies on theorem 1 and a result of Nakano applied to the families of hyperplanes tangent to a given variety. The techniques used in the proof of theorem 1 are then applied to the study of the adjunction mapping associated to a very ample divisor D on a surface S. This completes and unifies previous work of Sommese, Van de Ven and L. Ein. Let \(K_ S\) denote the canonical divisor on S. The linear system \(| D+K_ S|\) is base-point free except for some well-understood cases. Denote by \(S\to^{\alpha}\hat S\to^{\beta}{\mathbb{P}}^ n\) the Stein factorization of the morphism determined by \(…
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14B25
zbMATH DE Number: 3957276
zbMATH Keywords: vanishing theorem for cohomology
zbMATH Keywords: effective smooth divisor
zbMATH Keywords: morphism of divisor to curve
zbMATH Keywords: extension of divisor morphism
zbMATH Keywords: canonical divisor
MaRDI profile type: Publication
cites work: Q4175362
cites work: On threefolds with low sectional genus
cites work: Canonical models of surfaces of general type
cites work: Q3887589
cites work: Surfaces with a hyperelliptic hyperplane section
cites work: On Zariski problem
cites work: On the hyperplane section principle of Lefschetz
cites work: Q4143433
cites work: Q3318887
cites work: Lectures on Curves on an Algebraic Surface. (AM-59)
cites work: Q4402617
cites work: Special Linear Systems on Curves Lying on a K 3 Surface
cites work: Projective Models of K - 3 Surfaces
cites work: The adjunction mapping and hyperelliptic divisors on a surface.
cites work: Hyperplane sections of projective surfaces. I: The adjunction mapping
cites work: On the 2-connectedness of very ample divisors on a surface
cites work: The theorem of Riemann-Roch for high multiples of an effective divisor on an algebraic surface
OpenAlex ID: W2014927503
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description: scientific article; zbMATH DE number 3957276