5
$\begingroup$

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?

Improve this question
$\endgroup$
6
  • 2
    $\begingroup$ I believe this fails for every curve with $g\geq 2$. By Proposition 5.2(ii) of this -- arxiv.org/abs/math/0602644 -- the blowing up is Fano only if the maximal slope on $X$ of an invertible subsheaf of the normal bundle $N_{X/\mathbb{P}^n}$ is strictly less than the fraction $\text{deg}_X (T_{\mathbb{P}^n}|_X)/\text{rank}(N_{X/\mathbb{P}^n})$. If $T_X$ is negative, then $\text{deg}_X (N_{X/\mathbb{P}^n})$ is strictly larger than $\text{deg}_X(T_{\mathbb{P}^n}|_X)$. The fraction is smaller than the slope of $N_{X/\mathbb{P}^n}$, so this fails even in the semistable case. $\endgroup$
    – Jason Starr
    Commented Mar 15, 2017 at 13:30
  • $\begingroup$ Many thanks Jason! (I wonder if it is a coincidence that $g\ge 2$ means general type). $\endgroup$
    – aglearner
    Commented Mar 15, 2017 at 13:44
  • 1
    $\begingroup$ Actually, I see now that this is already a problem if $g(X)$ equals $1$. Then $\text{deg}_X(N_{X/\mathbb{P}^n})$ equals $\text{deg}_X(T_{\mathbb{P}^n}|X)$, so that the fraction equals the slope of $N_{X/\mathbb{P}^n}$. However, the necessary condition is that the maximal slope of an invertible sheaf is bounded below this slope, not just for $X$, but also for covers $C\to X$. As explained in Corollary 6.9, that is impossible. $\endgroup$
    – Jason Starr
    Commented Mar 15, 2017 at 14:09
  • 3
    $\begingroup$ @Jason: There must be something wrong. The blow-up of $\mathbb{P}^3$ along a curve of bidegree $(p,q)$, with $p,q\leq 3$, is Fano. $\endgroup$
    – abx
    Commented Mar 15, 2017 at 14:31
  • 1
    $\begingroup$ I corrected the argument in the answer below. The argument applies if $g(X) \gg 0$. $\endgroup$
    – Jason Starr
    Commented Mar 15, 2017 at 15:43

1 Answer 1

Reset to default
4
$\begingroup$

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

Improve this answer
$\endgroup$
2
  • $\begingroup$ Thank you for this answer! I have a question concerning lines -4-5 (I don't understand something). It looks to me that you claim that a blow up of $\mathbb P^5$ in any curve is not Fano? Is not the blow up in a line in $\mathbb P^n$ Fano for any $n$? (or the line is the only exception?) $\endgroup$
    – aglearner
    Commented Mar 15, 2017 at 16:07
  • 1
    $\begingroup$ I edited my answer to address your question. $\endgroup$
    – Jason Starr
    Commented Mar 15, 2017 at 16:28

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .