Actual and virtual dimension of codimension \(2\) general linear subspaces in \(\mathbb{P}^n\) (Q2665222)

The author studies the actual and virtual dimension of a spaces of hypersurfaces of given degree containing \(s\) codimension \(2\) general linear subspaces in \(\mathbb{P}^{n}_{\mathbb{K}}\), where \(\mathbb{K}\) is an algebraically closed field of characteristic \(0\). We start by recalling basics. Denote by \(R = \mathbb{K}[\mathbb{P}^{m}_{\mathbb{K}}] = \mathbb{K}[x_{0}, \dots, x_{n}]\) the homogeneous coordinate ring of \(\mathbb{P}^{n}_{\mathbb{K}}\). One considers \(s\) distinct linear subspaces \(\Lambda_{1}, \dots, \Lambda_{s} \subset \mathbb{P}^{n}_{\mathbb{K}}\) of codimension \(2\), and we denote by \(X = m_{1}\Lambda_{1} + \dots + m_{s}\Lambda_{s}\) the scheme defined by the ideal \[I_{X} = I(\Lambda_{1})^{m_{1}} + \dots + I(\Lambda_{s})^{m_{s}} \subset R\] which is generated by homogeneous polynomials vanishing on each \(\Lambda_{i}\) to order at least \(m_{i}\). The Hilbert function of \(X\) is the function \[\mathrm{HF}_{X} \, : \, \mathbb{N} \ni t \mapsto\dim_{\mathbb{K}}[R/I_{X}]_{t} \in \mathbb{N},\] where \([\cdot ]_{t}\) denotes the degree \(t\) part. It is well-known that there exists a polynomial \(\mathrm{HP}_{X} \in \mathbb{Q}[t]\), called the Hilbert polynomial of \(X\), such that \(\mathrm{HP}_{X}(t) = \mathrm{HF}_{X}(t)\) for all \(t\) sufficiently large. The actual dimension of \(X\) is the dimension of the vector space of forms in \(I_{X}\) of degree \(t\): \[\mathrm{adim}_{n}(X,t) = \dim_{\mathbb{K}}[I_{X}]_{t} = \dim_{\mathbb{K}}[R]_{t} - \mathrm{F}_{X}(t).\] The virtual dimension is defined as: \[\mathrm{vdim}_{n}(X,t) = \dim_{\mathbb{K}}[R]_{t} - \mathrm{HP}_{X}(t).\] We say that \(X\) admits an unexpected hypersurface of degree \(t\) if \(\mathrm{adim}_{n}(X,t) > 0\) and \[\mathrm{adim}_{n}(X,t) > \mathrm{vdim}_{n}(X,t).\] Now we are ready to present the first result of the paper. Theorem A. Let \(X = \Pi_{1} + \dots + \Pi_{s}\) be a subscheme of \(\mathbb{P}^{n}_{\mathbb{K}}\), with \(n\geq 2\) and \(s\geq 0\), of \(s\) general linear subspaces \(\Pi_{1}, \dots, \Pi_{s}\) of codimension \(2\).Then \(\mathrm{vdim}_{n}(X,t) = S_{n,s,t}\), with \[S_{n,s,t} = \sum_{i=0}^{N(n,s)} (-1)^{i} \binom{s}{i} \cdot \binom{t+n-2i}{n-2i},\] where for given \(n,s\) we take \(N(n,s) = \min \{ [n/s], s\}\). In the same setting, the author shows the following. Theorem B. Let \(\Pi_{1}, \dots,\Pi_{s}\) be general linear subspaces of codimension \(2\) in \(\mathbb{P}^{n}_{\mathbb{K}}\). Consider the scheme \(X = \Pi_{1} +\dots+\Pi_{s}\). If \(S_{n-2p,\, s-p, \, t} > 0\) for \(p \in \{1,\dots, N(n,s)-1\}\), then \[\mathrm{adim}_{n}(X,t) \geq \mathrm{vdim}_{n}(X,t).\] In particular, one has additionally that \[\mathrm{adim}_{n}(X,n+k) = \mathrm{vidm}_{n}(X,n+k) > 0 \text{ for } k\geq 3.\] From the perspective of unexpected hypersurfaces, the author proves the following result. Theorem C. Let \(X' = 3\Pi_{1} + \dots + 3\Pi_{n+1}\), whre \(n\geq 3\). Then \(X'\) admits an unexpected hypersurface of degree \(3n+1\).