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Let $X$ be a Fano 3-fold with terminal singularities. Is there some bound (possibly explicit) for the Picard rank of $X$ ?

If $X$ is smooth, it is well-known that the bound is $10$, obtained by del Pezzo fibrations of degree $1$.

With some assumptions on $X$, the bound is $7$ (Nikulin, "On the Picard number of Fano 3-folds with terminal singularities"). Does someone knows what happens in the general case?

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It might be good to say a bit about Nikulin's assumptions. Anyway, according to the paper the paper "The Defect of Fano 3-folds" by A.-S. Kaloghiros (arxiv 0711.2186), Namikawa proved that a terminal Gorenstein Fano 3-fold can be deformed to a smooth Fano with the same Picard rank. I don't have access to Namikawa's article, but maybe it's a good place to look for more results in this direction. – user5117 Jun 20 '12 at 15:05
The assumptions of Nikullin are that there is no small contraction, no contraction to a curve or to a surface (only divisorial contractions). If you have only Gorenstein singularities, the bound is also 10. See "On the Picard number of singular Fano varieties" by Gloria Della Nocce. In general, it is bounded because of the boundedness of Fano varieties with terminal singularities. See "Boundedness of canonical Q-Fano 3-folds" by Kollár, Miyaoka, Mori, Takagi. But it would be good to have an explicit bound, not so big. – Jérémy Blanc Jun 20 '12 at 15:30
If no answer - may be e-mail to Aleksandr Pukhlikov - you can find his mail from arXiv. He seems to me best expert in Russia in such kind of things after V. Iskovskix unfortunately passed – Alexander Chervov Jun 21 '12 at 10:45

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