Let $D$ be a divisor on a non-singular projective variety $X$.
If $D=Y_1+Y_2$ for two prime divisors $Y_1, Y_2$, then $D$ is smooth iff $Y_1$ does not intersect $Y_2$ and both are smooth?
My reasoning is that if $Y_1\cap Y_2\not=\emptyset$, then around the intersection we have a defining equation $f=f_1f_2$. But then using the Jacobi criterion for smoothness we find that at the intersection $\frac{\partial f}{\partial x_i}=0$ for all $i$, and hence $Y$ is not smooth.
In particular on $X=\mathbb{P}^n$ the only smooth sections of $\mathcal{O}_X(d)$ correspond to irreducible, smooth, homogeneous degree $d$ polynomials. Is that correct?