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In a previous question, it appears that a collection of points in $\mathbb P^2$ is in general position precisely when the blow-up of $\mathbb P^2$ at those points has no $(-2)$-curve. Is there a similar criterion which tells you when a collection of points in $\mathbb P^3$ is in general position in terms of the geometry of the blow-up of $\mathbb P^3$ at those points?

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    $\begingroup$ Could you try to make your question more specific? It is hard to know what you are interested in. $\endgroup$ Mar 20, 2011 at 0:01
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    $\begingroup$ I don't have an answer, but I can at least provide a guess at the definition of "general position." Degree $d$ hypersurfaces in $\mathbb P^3$ form a projective space of dimension $\frac{(d+1)(d+2)(d+3)}{6}-1$. The degree $d$ hypersurfaces containing a given point form a hyperplane in that projective space. We say that $m$ points are in general position if the intersection of those $m$ hyperplanes has the right dimension, $\frac{(d+1)(d+2)(d+3)}{6}-1-m$, (or is empty if this number is negative) for every $d$. $\endgroup$ Mar 20, 2011 at 1:56
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    $\begingroup$ It is not true that a collection of points in P^2 is in general position precisely when there is no (-2)-curve on the blowup. For instance if 9 points are the base locus of a pencil of cubics, there need not be such a curve. $\endgroup$ Mar 20, 2011 at 2:10
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    $\begingroup$ +Jack's comment: in the accepted answer to the question you (Sohail) quote Francesco mentions the exact same example that Jack does to support the claim that the description by the existence of $(-2)$ curves is only valid for a small number of points. Perhaps you should restrict the question to blow ups that still have a big anticanonical divisor. The analogous condition to the existence of $(-2)$ curves in higher dimensions would be that $-K_X$ is nef and big but not ample. $\endgroup$ Mar 20, 2011 at 5:12
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    $\begingroup$ A comment on terminology: as far as I know, being "in general position" should usually be a condition that holds on a Zariski-open subset of the configuration space. (More precisely, we say that a property P of tuples of points holds for tuples in general position if it holds for all tuples in some Zariski-open subset U_P of the configuration space.) Anton's definition seems to impose a countably infinite set of open conditions on tuples. In such a case one usually speaks of tuples "in very general position". $\endgroup$
    – user5117
    Mar 22, 2011 at 10:10

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