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On connected divisors. (English) Zbl 1019.14003
From the introduction: Before the coming of $$\mathbb{Q}$$-divisors and of the Kawamata-Viehweg theorem, E. Bombieri [Publ. Math., Inst. Hautes Étud. Sci. 42, 171-219 (1972; Zbl 0259.14005)] noted a vanishing theorem, to whose effect $$h^1({\mathcal I}_D)=0$$, for any numerically connected divisor $$D$$ on a smooth surface $$S$$, with $$D^2>0$$. In the case of surfaces, it makes sense to strengthen the notion of numerical connectedness into that of $$k$$-connectedness, introduced by Bombieri [loc. cit.], $$k$$ being a measure of how connected the divisor $$D$$ is. A few years later, A. van de Ven [Duke Math. J. 46, 403-407 (1979; Zbl 0458.14003)] proved that every very ample divisor on a surface is 2-connected, with only two exceptions.
Nowadays the Kawamata-Viehweg theorem gives much stronger vanishings, but they come at a price: The divisor $$D$$ must be nef and big; also, the proof requires the full force of $$\mathbb{Q}$$-divisor techniques.
In the present paper, we generalize both Bombieri’s and van de Ven’s theorem by using a more down-to-earth approach. Indeed we prove that, for a numerically connected divisor $$D$$ on a smooth $$n$$-dimensional variety $$X$$, $$h^1({\mathcal I}_D)=0$$, provided that $$D^n>0$$ and $$h^0(D)\geq 3$$. Subsequently, we introduce the notion of $$k$$-connected divisors for higher dimensional varieties, which reduce to Bombieri’s in the case of surfaces. Equipped with this definition, we prove that every very ample divisor on a smooth threefold is 3-connected, but for a finite number of exceptions, which are completely described.
##### MSC:
 14C20 Divisors, linear systems, invertible sheaves 14F17 Vanishing theorems in algebraic geometry 14J30 $$3$$-folds
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