ag.algebraic geometry – Open problems and examples of special linear systems


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
$$
vdim L(d,m) = {d+2choose 2} – sum {m_i+1choose 2} -1.
$$

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.