Let $X$ be a complex Fano manifold such that each extremal ray of $\overline{\text{NE}(X)}_{\mathbb{R}}$ is generated by a primitive class in $H_2(X;\mathbb{Z})$ of a free rational curve. Thus, the extremal rays are all fiber type. (The "primitive" hypothesis rules out, e.g., conic bundles where "half" of the fiber class is integral.)

Question. Is the cone of effective curves in $H_2(X;\mathbb{Z})$ a free $\mathbb{Z}_{\geq 0}$-semigroup, $\mathbb{Z}_{\geq 0}^r$?

There are many positive examples, e.g., Fano manifolds with a transitive (algebraic) action of a complex Lie group and complete intersections in these of low degree. However, for the general question, the best results that I have found are in the following article of Jaroslaw Wisniewski.

MR1639552 (2000e:14018)
Wiśniewski, Jarosław A.
Cohomological invariants of complex manifolds coming from extremal rays.
Asian J. Math. 2 (1998), no. 2, 289–301.

  • $\begingroup$ Is it true that the tangent bundle $T_X$ is nef under this assumption? $\endgroup$ Nov 12 '18 at 21:17
  • $\begingroup$ @JonathanFrink. No, that is not true. A Fano hypersurface in projective space of Fano index $2$ (and dimension $\geq 3$) contains lines on which the restriction of the tangent bundle has a negative summand, e.g., lines of "type II" in cubic threefolds. $\endgroup$ Nov 12 '18 at 21:23

I think that this is open in general, and that one would expect the answer to be positive. Related references are:

  • MR1103910 Wiśniewski, Jarosław A., On contractions of extremal rays of Fano manifolds. J. Reine Angew. Math. 417 (1991), 141–157

He shows in Theorem 2.2 that if X is a smooth Fano where every extremal ray is of fiber type, then the number of extremal rays is at most dim(X), hence also rho(X) is at most dim(X), where rho(X) is the Picard number

  • MR3533196 Druel, Stéphane, On Fano varieties whose effective divisors are numerically eventually free. Math. Res. Lett. 23 (2016), no. 3, 771–804

He classifies the cases where rho(X)=dim(X), and NE(X) turns out to be simplicial.

  • Ou, Wenhao, Fano varieties with Nef(X)=Psef(X) and rho(X)=dim(X)-1, Manuscripta Math. 2018

He studies the next case rho(X)=dim(X)-1, I think that probably NE(X) turns out to be simplicial too, but I did not check

  • MR2474316 by myself, Quasi-elementary contractions of Fano manifolds. Compos. Math. 144 (2008), no. 6, 1429–1460

It follows from Theorem 4.1 that NE(X) is always simplicial if dim(X) is at most 5.

I hope this helps. Please let me know if you find more general results, I am interested in this question!


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