I have a reducible projective variety over the $\mathbb{C}$ of dimension 3. The irreducible components of the variety intersect normally. Is there a way to test ampleness of a divisor on such a variety?

I know the Nakai criterion but i do not know how to apply it in this special case.

  • 2
    $\begingroup$ For sure it is enough to check ampleness of your divisor restricted to each irreducible component, but probably you already know this. $\endgroup$ – Gianni Bello Dec 7 '10 at 20:12
  • $\begingroup$ It depends on what extra information you have about your 3-fold. Do you know the Picard group (or anything about the vanishing of cohomology on the 3-fold)? Or the cone of curves? If you know the cone of curves then you could just use the Nakai-moishezon on each component. $\endgroup$ – J.C. Ottem Dec 7 '10 at 20:28

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