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Sur le cone des 1-cycles effectifs en dimension 3. (On the cone of effective 1-cycles in dimension 3). (French) Zbl 0589.14037
Let X be a normal projective threefold over the complex numbers. Assume X has only canonical singularities in the sense of M. Reid [Journées géom. Algébr., Angers/France, 273-310 (1980; Zbl 0451.14014)]. Such threefolds have a canonical Q-divisor, denoted by K. Let NS(X) be the Néron-Severi group and NE(X) the cone generated by effective curves in the dual of NS(X)$$\otimes {\mathbb{R}}$$. - Let $$NK(X,0)=\{Z\in \overline{NE(X)}| \quad K.Z<0\}.$$ The author defines a face F of NK(X,0) to be extremal if $$\bar F\subset NK(X,0)$$, also F to be minimal if the number of irreducible curves whose class lies in F is finite. Extremal rays were studied in depth by S. Mori [Ann. Math., II. Ser. 116, 133-176 (1982; Zbl 0557.14021)].
The author proves the following theorem: ”Let X and K be as above and F a minimal extremal face of NK(X,0). Then for every curve C such that this class of C belongs to F the following holds: $$(i)\quad C\quad is$$ a smooth rational curve; $$(ii)\quad -1<KC<0.''$$- As a corollary, he obtains the following: ”Let X be a normal Gorenstein projective threefold with nonnegative Kodaira dimension. Assume that all extremal rays of X are minimal. Then the canonical divisor of X is numerically positive”.
The author also remarks that any smooth threefold with non-negative Kodaira dimension has a birational model which has only canonical singularities and satisfying the hypothesis above on extremal rays. (This is a consequence of theorems of M. Reid and Y. Kawamata.) If in addition they were Gorenstein, the corollary would be applicable but unfortunately they are not in general. Some of the intermediate propositions should be of independent interest.
Reviewer: N.Mohan Kumar

##### MSC:
 14J30 $$3$$-folds 14C20 Divisors, linear systems, invertible sheaves
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##### References:
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