Extension of morphisms defined on a divisor
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 \(| D+K_ S|\), and suppose that dim \(\beta(\hat S)=2\). The author proves that \(\beta\) is an embedding provided \(D^ 2\geq 10\). Furthermore, if \(D^ 2=9\) and \(\beta\) fails to be an embedding, then D is numerically equivalent to 3V, for some divisor V.
scientific article; zbMATH DE number 3957276
scientific article