Suppose $X$ is a smooth toric Fano $3$-fold, and $D$ is a torus invariant divisor corresponding to a face of the polytope associated to $X$. I would like to search for (smooth) curves $C \subset D$, such that $Bl_{C}(X)$ is Fano.
Given the explicit descriptions of $Pic(X)$ and $Pic(D)$ in terms of boundary divisors (note that $D$ is a toric surface), and the description of $-K_{X}$ as the sum of the boundary divisors, I hope that there is a systematic way to search for such classes $[C] = \alpha \in Pic(D)$, or at least some observation/restriction that reduces the search.
So far, the only restriction I was able to be able to (maybe) come up with was
$$-K_{X} \cdot C_{j} > C \cdot C_{j},$$
for each torus invariant boundary divisor $[C_{j}] \in Pic(D)$ (the second intersection being taken in $D$). However, my justification for this is a bit sketchy. If this turns out to be true then I would be interested to know non-example which satisfy these restrictions (i.e. C as above such that $Bl_{C}(X)$ is not Fano).
Note that $C$ need not be rational as the example of a plane cubic in $\mathbb{P}^{3}$ shows.
Apologies for the vagueness of the question, any remarks would be greatly appreciated.
