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Context: Consider a set of points $p_1,\cdots,p_r$ in $\mathbb{P}^2$ and let $X$ be its blowup at these points. Denote the hyperplane class in $\mathbb{P}^2$ by $H$ and the exceptional divisors by $E_i$ for $i=1,\dots,r$.

Given positive integers $d$ and $m_1,\dots,m_r$ define $L(d,m)$ to be the linear system of all degree $d$ forms vanishing at the point $p_i$ with multiplicity at least $m_i$. In other words, $L(d,m) = (H^0(X,dH-\sum m_iE_i)-\{0\})/\mathbb{C}^\times$.

The virtual dimension of $L(d,m)$ is defined as $$ vdim L(d,m) = {d+2\choose 2} - \sum {m_i+1\choose 2} -1. $$ Essentially, the first term is the degrees of freedom and the latter the constraints imposed by the vanishings.

Then, the expected dimension of $L(d,m)$ is $$edim L(d,m)= \max\{vdim L(d,m),-1\}.$$

It's known that $dim L(d,m)\geq edim L(d,m)$ and we say the linear system is special if this inequality is strict.

Question 1: Are there known examples of linear systems where $vdim L(d,m)<-1$ but $dim L(d,m)>0$? How about the case $m_i=1$?

A similar problem would be the following:

Question 2: Are there known examples where $dim L(d,m) - edim L(d,m) > k$ for some $k>1$?

In general I'm interested in knowing what is the current state of this kind of problem. How active is it as a line of research and if there are any open conjectures along these lines. I'd also appreciate if someone could share some known examples and references.

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    $\begingroup$ Q1: take $p_1,\ldots , p_r$ on a line, with $r\geq 6$, and $d=2$. Then $\operatorname{vdim} L(d,m) =5-r$ and $\operatorname{dim} L(d,m) =2$. In the same way you can easily cook up an answer to Q2. $\endgroup$ – abx Sep 30 '20 at 5:57
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    $\begingroup$ By the way, this question has been much studied in algebraic geometry. You can look for instance at this paper and the references given there. $\endgroup$ – abx Sep 30 '20 at 5:59
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    $\begingroup$ Here's another reference. $\endgroup$ – Zach Teitler Sep 30 '20 at 6:47

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