# 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.