Fano manifold admit an smooth anti-canonical divisor?

Question

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

Answer 1

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.

Answer 2

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.