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Let $X$ be the blowing-up of $\mathbb{P}^n$ in $r$ points in general position. Let $\{H,E_1,...,E_r\}$ be the standard basis of $Pic(X)$. Further let $D=dH-\sum m_j E_j$, be an effective divisor, with $m_j \in \mathbb{Z}$. Is it true that if $m_1\geq 0$ then $E_1 \not\subseteq Bs(|D|)$?

If we suppose all the $m_i$'s are non negative, this corresponds to say that $dim |dH-m_1E_i...-m_rE_r|>dim |dH-(m_1+1)E_1-...-m_rE_r|$. In other words this means that if we consider all the hypersurfaces of $\mathbb{P}^n$ of degree $d$ passing through $r$ general points $\{p_1,...,p_r\}$ with multiplicities $m_1,...,m_r$, then they do not all pass through $p_1$ with multiplicity $m_1+1$. My intution brings me to say this is obvious but I cannot prove it. In fact everything is trivial for nonspecial linear systems, but, a priori, special linear systems may be a problem.

Note that the generality of the points is necessary: for example you can simply take $\mathbb{P}^2$ blown up in 3 collinear points and note that $E_1 \subseteq Bs(|H-E_2-E_3|)$.

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What is the relationship between $d$ and $r$? If $d$ is large enough with respect to $r$, then what you want is true; otherwise, the stable base locus could be the whole $X$, since there may not be non-zero section of the relevant line bundle. –  damiano Oct 1 '10 at 9:38
    
He is assuming that the divisor is effective. –  Angelo Oct 1 '10 at 9:44
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In $P^2$, this would follow from the famous Segre-Harbourne-Gimigliano-Hirschowitz conjecture (see ams.org/notices/199902/miranda.pdf for instance) which describes all special systems. So it will be hard to come up with a counterexample ;) But it does not seem an easy thing to prove without SHGH.. –  quim Oct 1 '10 at 14:00
    
The question seems to be open: see section 3.9 in arXiv:1101.4363 –  quim Mar 30 '11 at 11:47
    
@Anyone who cares: I only edited formatting, not content. (On my screen the last bunch of math symbols jumped up over the first line and I made that go away.) Cheers. –  Sándor Kovács Mar 31 '11 at 2:37
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2 Answers

It seems to me that my question is easier than SHGH conjecture. I think that prop. 2.3 in "Weakly defective varieties" by Chiantini-Ciliberto is a positive answer to my question.

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This is correct. I just saw this question and was going to point out this result, but you beat me to it. –  Jack Huizenga Mar 31 '11 at 3:02
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In the same vein as in a comment above, I'll comment when the base locus gets multiplicity. In such a case, this is equivalent to the Harbourne-Gimigliano-Hirschowitz conjecture which more or less reads as follows for $\mathbb P^2$:

a linear system $L$ is special iff it contains a multiple $(-1)$-curve in its base locus.

Where $L=\mathcal O(dH-\sum m_ip_i)$ and special means $h^1(L)\neq 0$. Here is interesting point is that all the points are in general position and even so, they might very well generate a special linear system. Besides the part that reads "multiple" is important. Here is an example where one has a $(-1)$-curve in the base locus and the linear system is NOT special,

all the plane cubics having two double points at $p_1, p_2$

Here, such cubics are all of the form $QL$ where $L$ is the line $p_1p_2$. Now, $L$ is a $(-1)$-curve and it is in the base locus. However the linear system is not special.

By the way, apparently example to keep in mind of what goes wrong with these special linear systems is of the following type.

Plane quartics having 5 double points.

The space of quartics has dimension 14 and 5 double points impose 15 conditions, therefore this linear system is expected to be empty. However take a conic $Q$ passing through those points and double it $Q^2$ (here the multiple of the curve showed up). This is quartic having 5 double points as we asked. This says that neither the general position is important here nor the system is empty. The conjecture claims that that multiple $(-1)$-curve characterizes all the special linear systems!

S. Yang solved this conjecture for $\mathbb P^2$ and multiplicity of the points less than 7.

Alexander-Hirschowitz solved the case for $\mathbb P^n$ and multiplicity 2. Here is a reference

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@Csar: The SHGH conjecture would give an answer to the original question, and it seems hard to get an answer without SHGH, but it is not clear to me whether SHGH is necessary in order to answer this. Are you claiming both questions are equivalent? Besides, the bound on the multiplicity for wich the conjecture is known in the plane is now 11, see Dumnicki-Jarnicki arXiv:math/0505183 –  quim Mar 30 '11 at 8:59
    
No, I don't claim the two questions are equivalent, I stated that when the the base locus has multiplicity the question is equivalent to the SHGH conjecture. Actually, I gave an example when the base locus is a (-1) curve but the linear system is not special. Though my answer seems to be confusing, I'll try to make clearer. –  Csar Lozano Huerta Mar 31 '11 at 3:08
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