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



Fano manifold admit an smooth anti-canonical divisor?
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*](http://www.google.it/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=0CCMQFjAA&url=http%3A%2F%2Fwww.maths.ed.ac.uk%2Fcheltsov%2Fshokurov%2Fpdf%2Felephant.pdf&ei=f0LKVOPtF9fvaO65gYgC&usg=AFQjCNFPX_liED_da0wBLoycMKGuf04Amg&bvm=bv.84607526,d.d2s), 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*. 


