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

• 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. – abx Sep 30 '20 at 5:57
• 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. – abx Sep 30 '20 at 5:59
• Here's another reference. – Zach Teitler Sep 30 '20 at 6:47