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Let $X^n_r$ the blow-up of $\mathbb{P}^n$ at $r$ points in very general position.

(1) It is known in general the Effective Cone or the Numerical Effective Cone of those algebraic variety?

Let $X$ be an algebraic variety.

(2) Where i can find equivalent conditions for a divisor class $D\in Pic (X)$ to be effective?

Thanks in advance !

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    $\begingroup$ Probably you know the following. For $n=2$ and $r \leq 8$, one knows explicitly the exceptional curves (Manin, Cubic Forms, Theorem 26.2). Thus one can can check explicitly if a divisor is nef, by intersecting with the exceptional curves. Then for surfaces, the nef cone is dual to the effective cone, so one can explicitly check if a divisor is effective (cont)... $\endgroup$ Sep 29, 2013 at 17:32
  • $\begingroup$ ...(cont) I suspect when $X_r^n$ is Fano, one can similarly explicitly describe the nef cone (following the cone theorem), however I'm not sure how helpful that is. For $r \geq 9$ and $n=2$, the nef cone isn't known explicitly (this is related to Nagata's conjecture). The point (I think) is $X$ is no longer Fano, and there are infinitely many exceptional curves, as $9$ points determine a cubic. $\endgroup$ Sep 29, 2013 at 17:33
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    $\begingroup$ @Ruadhaí Dervan --- there is a description of the cone when $r=9$ and $n=2$, though maybe it's a matter for debate whether it's "explicit" or not: the cone is spanned by the $(-1)$-classes, together with the class of $-K$. I think this is originally due to Borcea, in "On desingularized Horrocks--Mumford quintics". For $r \geq 10$, the best results I'm aware of are due to de Fernex, but there is no complete description yet. $\endgroup$
    – user5117
    Sep 30, 2013 at 12:38

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As explained in the comment of Ruadhai Dervan, the case where $X$ is not Fano (or even worse, when $X$ is not weak-Fano) is probably hard to deal with.

The Picard group of $X_r^n$ is generated by $H$, the pull-back of a hyperplane, and by $E_1,\dots,E_r$, the divisors associated to the points blown-up.

In dimension $2$, Nagata's conjecture corresponds to say that if $r> 9$ and $dH-\sum_{i=1}^r a_iE_i$ is effective, then $\sum a_i< \frac{d}{\sqrt{r}}$ (the result is false for $r\le 8$, which corresponds to the Fano case and for $r=9$, as you take a cubic through the points). This is true when $r$ is a square, and proved by Nagata, but still open if $r$ is not a square and $r>9$. There are however partial results in the direction of the conjecture, that you can find on the web by looking for "Nagata conjecture for curves".

In dimension higher, I don't know what are the results when the variety is not Fano. You can also describe when $X_r^n$ is Fano, or only weak-Fano, depending on $r$ and $n$.

In "Weak Fano threefolds obtained by blowing-up a space curve and construction of Sarkisov links." Proc. Lond. Math. Soc. (2012) 105(5): 1047-1075, S. Lamy and myself considered the case of blow-ups of curves in $\mathbb{P}^3$, but also did the easier case of points (see section 2.7), which is basically an exercise.

I summarise the result: For $n\ge 3$, $X_r^n$ is Fano if and only if $r=1$ and is weak-Fano (and not Fano) if and only if $n=3$ and $2\le r\le 7$.

You can also have a look at the texte and see the conditions that you need to put on the points so that you obtain a weak-Fano $3$-fold.

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The nef cone of $X_r^n$ is particularly well behaved when $X_r^n$ is a Mori dream space. In this case it has a nice decomposition made of the nef cones coming from its small $\mathbb{Q}$-factorial modifications. It is known that.

  • $X_{r}^n$ for $n\geq 5$ is a Mori dream space if and only if $r\leq n+3$,
  • This bound can be improved for $n=3,4$. Indeed $X_r^3$ is a Mori dream space if and only if $r\leq 7$, and $X_r^4$ is a Mori dream space if and only if $r\leq 8$,
  • finally $X_r^2$ is a Mori dream space if and only if $r\leq 8$ (these are Del Pezzo surfaces).

You can find this here:

http://arxiv.org/abs/math/0505337

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