Let Y be a smooth cubic threefold and p a point. Is the blowup $X=Bl_p Y$ a weak Fano (in the sense that $-K_X$ big and NEF)?
On page 2 of https://arxiv.org/pdf/1409.7778.pdf, this seems to be asserted. However I'm curious about a reference and how this is proven?
The projection $\pi_p:Y\subset\mathbb{P}^4\dashrightarrow \mathbb{P}^3$ with center $p\in Y$ induces a morphism $f:X\rightarrow\mathbb{P}^3$. Note that if you call $H$ the pull-back on $X$ of the hyperplane section of $\mathbb{P}^4$ and $E$ the exceptional divisor then $f$ is induced by the divisor $D = H-E$.
Therefore $D = H-E$ is base point free and in particular nef. Note that $H-E$ is note ample because $f$ contracts the strict transforms of the lines in $Y$ through $p$ (there is always a line in $Y$ through $p$).
The morphism $f$ is generically $2$ to $1$, hence $D$ is big as well.
Now by adjunction $-K_Y \cong \mathcal{O}_{Y}(5-3) = \mathcal{O}_Y(2)$. Then $$-K_X = 2H-2E = 2D$$ is big and nef but not ample. This means that $X$ is weak Fano but not Fano.
If you call $X_k$ the blow-up of $Y$ at $k$ general points, assuming that $Pic(Y)\cong \mathbb{Z}[H]$ then by Theorem 1.2 here
https://arxiv.org/pdf/1705.04972.pdf
$X_k$ is Weak Fano if and only if $k\leq 2$.
