simple normal crossing divisors on Fano manifold

Question

Let $M$ be a Fano manifold. And $D=\mathop\sum\limits_{i=1}^r\tau_iD_i\in|-\lambda K_M|$ is a simple normal crossing $\mathbb{R}$-divisor where $\tau_i\in(0,1)$. Can we know that $D_i$s are ample (or $c_1(D_i)\geq 0$ for $i=1,\cdots,r$)? If not, for $D\in |-\lambda K_M|$, can we find such a decomposition of $D$ so that the condition "$c_1(D_i)\geq 0$ for $i=1,\cdots,r$" satisfies?

What if $M$ is a toric Fano manifold and $D_i$ are toric divisors corresponding to the faces of the moment polytope?

Answer 1

Let $M=M_1\times M_2$ and $D_i$ the pull back of an anti-canonical divisor on $M_i$. Then $D_1+D_2\in |-K_M|$ but they are individually not ample. (They are nef though).

I don't know what you $\lambda$ is, but if you want coefficients strictly between $0$ and $1$, then choose two different anti-canonical divisor on each factor and multiply them by $1/2$.

To get an example with not nef components take $M$ to be the blow-up of $\mathbb P^n$ in a single point with $E\simeq \mathbb P^{n-1}$ the exceptional divisor. Then $M$ is a $\mathbb P^1$-bundle on $\mathbb P^{n-1}$. (There is a projection $\pi:M\to E$ with fibers isomorphic to $\mathbb P^1$).

Then the anti-canonical divisor of $M$ is something nef pulled-back from the $\mathbb P^{n-1}$ plus a multiple of $E$ which is not nef. More precisely, one can write $K_M\sim a\pi^*H+bE$ where $H\subseteq E$ is the hyperplane class. Now, if $F\simeq \mathbb P^1$ is a fiber of $\pi$, then $K_M|_{F}\sim K_{\mathbb P^1}$ so $b=-2$ and $(K_M+E)|_E\sim K_E\sim -nH$ and $E|_E\sim -H$, so $a=-(n+1)$, that is, $$ -K_M \sim (n+1)\pi^*H +2 E. $$

You can also consider the blow-up morphism $\sigma: M\to \mathbb P^n$ and write $$ -K_M \sim \sigma^*(-K_{\mathbb P^n}) - (n-1) E $$ in which case $-(n-1)E$ is not nef, but then the coefficients are not in the range you declared.