The study of spaces of forms of given degree having points of fixed multiplicity is quite classical and it appears in relation to many other problems. Let \(L = {\mathcal L}_n(d,m_1,\dots ,m_r)\) be the linear system of degree \(d\) hypersurfaces passing through \(r\) general points \(p_1,\dots ,p_r\) in \({\mathbb P}^n\) with multiplicity \(m_i\) at each \(p_i\). The virtual dimension vdim\(L\) is defined as vdim\(L := {d+n \choose n}-\sum_{j=1}^r {m_j+n-1\choose n}\), and the expected dimension of \(L\) is edim\(L = \max\, \text{vdim}L, -1 \}\); it is of interest to know when the actual dimension of \(L\) is bigger than edim\(L\) (in this case we say that \(L\) is \(special\)).
Except for \(n=2\) or \(d=2\), very little is known about this problem (for \(d=2\), all special system are described by a theorem of Alexander and Hirschowitz, while for \(n=2\) there is a conjecture which describes the special systems, and it has been proved when \(d\), \(r\) or the \(m_i\)’s are “small enough”, but it is still open in the general case).
If we let \(\alpha_L=\min \{d\in {\mathbb N}| L \neq \emptyset \}\); \(\tau_L=\min \{d\in {\mathbb N}| L \neq \emptyset \wedge L\;\text{is\;non-special}\}\), then in order to study the speciality of \(L\) it is of interest to have bounds on \(\alpha_L\) and \(\tau_L\).
The main result in this paper is the following:
Let \(n\geq 2\) and \(d,k,m_1,\dots ,m_s,m_{s+1},\dots ,m_r \in {\mathbb N}\); if \(L_1 = {\mathcal L}_n(k,m_1,\dots ,m_s)\) and \(L_2 = {\mathcal L}_n(d,m_{s+1},\dots ,m_r,k+1)\) are non-special, and \((\text{vdim}L_1+1)(\text{vdim}L_2+1)\geq 0\), then \(L = {\mathcal L}_n(d,m_1,\dots ,m_r)\) is non-special.
Based on this result, algorithms are given which bound \(\alpha_L, \tau_L\) for many linear systems \(L\).