Let $X$ be a smooth complex variety. Is it always possible to find an embedding $\varphi: X\to \mathbb CP^n$ for some $n$, such that the blow up of $\mathbb CP^n$ at $\varphi(X)$ is a Fano variety? 


Let us call the above class of varieties $F$-embeddable. How large is this class? How large is its complement?



Fano blow ups of $\mathbb CP^n$
**Note.** My comment above was wrong; I had the wrong denominators. When you correct the denominators, the formula gives an asymptotic result. 


In my comment I wrote the wrong formula for the denominator of that fraction. The correct statement is that the blowing up of $\mathbb{P}^n$ along a smooth subvariety $X$ of pure dimension $d$ is Fano if and only if there exists $\epsilon>0$ such that for every irreducible curve $B$ and for every nonconstant morphism $u:B\to \mathbb{P}^n$ both, $$\bullet \ \ \text{deg}\_B (g^{-1}(X)) \leq \frac{1}{c-1}\left(\text{deg}\_B(g^\*T\_{\mathbb{P}^n}) - \epsilon \ \text{deg}\_B (g^\*H)\right)\ \text{ if } g(B)\not\subset X, $$ $$\bullet \ \ \mu\_B^1(g^\*N\_{X/\mathbb{P}^n}) \leq \frac{1}{c-1}\left( \text{deg}\_B(g^\*T\_{\mathbb{P}^n}) - \epsilon \ \text{deg}\_B (g^\*H) \right)\ \text{ if } g(B)\subset X,$$
where $c$ equals the codimension $n-d$, and where $\mu^1\_B(g^\*N\_{X/\mathbb{P}^n})$ denotes the maximal slope of an invertible subsheaf of $g^\*N\_{X/\mathbb{P}^n}$. In characteristic $0$, up to replacing $B$ by a cover and replacing $\epsilon$ by a smaller positive number such as $\epsilon/2$, we may replace $\mu^1\_B$ by the maximal slope of any locally free subsheaf of $g^\*N\_{X/\mathbb{P}^n}$ of positive rank, cf. Corollary 6.9 of the link above.


**Claim.** In characteristic $0$, for every $d$, there are only finitely many deformation types of $F$-embeddable varieties of dimension $d$.


**Proof**
If $X$ is a linear subvariety, then the blowing up is Fano. Thus assume that $X$ is not a linear subvariety, e.g., this holds if $X$ is not abstractly isomorphic to $\mathbb{P}^d$. 


In that case, there exists a $2$-secant line $B$ to $X$ that is not contained in $X$. Then $\text{deg}\_B(g^{-1}(X))$ equals $2$, whereas the fraction equals $(n+1-\epsilon)/(n-d-1)$. This can only be Fano if $n\leq 2d+2$. Thus, for $n\geq 2d+3$, the blowing up is not Fano. In characteristic $0$, Kollár-Miyaoka-Mori proved that there are only finitely many deformation types of Fano manifolds of dimension $\leq n\_0$ for each $n\_0$. Setting $n\_0=2d+2$, it follows that for every integer $d>0$, there are only finitely many deformation types of smooth projective variety of dimension $d$ that admit a closed immersion $\phi:X\to \mathbb{P}^n$ for which the blowing up is Fano. **QED**


