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Given a smooth Fano variety $X$ over $\mathbb{C}$, we can define the index, $I(X)$, as the divisibility of $-K_X$ inside of $Pic(X)$. There is a theorem which states that $I(X) \leq n+1$, where $n$ is the dimension of the variety. Moreover varieties of index $n$ are quadrics and varieties of index $n+1$ are projective spaces. I'm curious about the following modified definition of the index $I'(X)$. Namely, define $I'(X)$ to be the maximal number $d$ such that $-K_X$ can be decomposed as a sum:

$$-K_X= D_1 + D_2 + \cdots + D_d $$

where $D_i$ are all ample. For Picard rank one, this of course doesn't do anything. Are there interesting examples of $I'(X) \geq n$ in higher Picard rank?

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    $\begingroup$ Funny question. A small remark: if $X$ is a surface, then $I(X)=I'(X)$. For $\mathbb{P}^2$ and $\mathbb{P}^1\times \mathbb{P}^1$ it is an easy check, and for the other surfaces, you have a $(-1)$-curve, whose intersection with $-K_X$ is $1$, so $I'(X)=1$. $\endgroup$ – Jérémy Blanc Aug 19 '13 at 10:17
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Keyword: "pseudo-index". See, for instance, the work of Jiung-Cheng Chen and Cho - Miyaoka - Shepherd-Barron.

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    $\begingroup$ Just to add a bit more to Jason Starr's excellent answer. The psuedo-index is the minimum number $-K_X \cdot C$ where $C$ is a rational curves. The papers above prove that pseudo index $\leq n+1$. Furthermore, Wisnieski proved that if the pseudo index is bigger than $n+2/2$, $H^2(X,\mathbb{R})$ is one dimensional. This seems to rule out interesting new examples. $\endgroup$ – Daniel Pomerleano Aug 19 '13 at 15:12

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