# Fano manifold admit an smooth anti-canonical divisor?

Let $M$ ba a compact Kaehler Fano manifold. Under which conditions $M$ admit a smooth anti-canonical divisor $D$

If $M$ is smooth and $\dim M =3$, then the answer is always. This is a theorem due to Shokurov, see

V.V. Shokurov: Smoothness of a general anticanonical divisor on a Fano variety, Math. USSR, Izv 14, 395-405 (1980).

In fact, Shokurov proves the following more precise result.

Theorem. Let $M$ be a smooth Fano threefold of index $r$ and let $H \in \textrm{Pic}(M)$ such that $rH \cong -K_M$. Then the general element of the linear system $|H|$ is smooth.

Using a terminology due (I think) to M. Reid, smooth anticanonical divisors in Fano threefolds are sometimes called elephants.

In general there may not be smooth anticanonical divisors. Of course if you take divisors in the linear system $|-mK_X|$ for $\gg 1$ there will be lots of smooth divisors, by Bertini's theorem (precisely when $|-mK_X|$ is basepoint free). But if you restrict to $m=1$ this is not always the case.

For an example of a Fano manifold without a smooth anticanonical divisor, see the work of Taro Sano: http://arxiv.org/abs/1302.0705 Example 2.9. Apparently the first such examples are due to Hoering-Voisin: http://arxiv.org/abs/1009.2853 Example 2.12.