Linear systems on the blow-up of \((\mathbb{P}^1)^n\) (Q5962501)

The authors study linear systems on \((\mathbb{P}^{1})^{n}\). It is a classical question to ask about the dimension of a linear system of hypersurfaces in \(\mathbb{P}^{n}\) of a given degree passing through finitely many very general points with prescribed multiplicities. It is well-known that for \(n=2\) the Segre-Harbourne-Gimiliano-Hirschowitz conjecture predicts the dimension of such linear systems. It is natural to ask about possible generalizations of the SHGH conjecture for other values of \(n\) or another varieties.NEWLINENEWLINEDenote by \(\mathcal{L} = \mathcal{L}_{(d_{1}, \dots, d_{n})}(m_{1}, \dots, m_{r})\) the linear system of hypersurfaces in \((\mathbb{P}^{1})^{n}\) of degree \((d_{1}, \dots, d_{n})\) passing through \(r\) very general points \(q_{1}, \dots, q_{r}\) with multiplicities \(m_{1}, \dots, m_{r}\). \textit{The virtual dimension} of \(\mathcal{L}\) is NEWLINE\[NEWLINE\text{vdim}(\mathcal{L}) = \prod_{i=1}^{n}(d_{i}+1) - \sum_{i=1}^{r}{ n + m_{i}-1 \choose n} - 1.NEWLINE\]NEWLINE Then the expected dimension of \(\mathcal{L}\) is defined as \(\text{edim}(\mathcal{L}) = \max \{\text{vdim}(\mathcal{L}), -1 \}\). The inequality \(\text{dim}(\mathcal{L}) \geq \text{edim}(\mathcal{L})\) always holds. If \(\text{dim}(\mathcal{L}) > \text{edim}(\mathcal{L})\), then we say that \(\mathcal{L}\) is special.NEWLINENEWLINEFor a given subset \(I \subset\{1, \dots, n\}\) we denoteNEWLINENEWLINENEWLINE\[NEWLINEP_{I}: (\mathbb{P}^{1})^{n} \ni ([x_{1}:y_{1}], \dots, [x_{n}:y_{n}]) \mapsto ([x_{i}:y_{i}] \, : \, i \in I)\in (\mathbb{P}^{1})^{|I|}.NEWLINE\]NEWLINE Moreover, we denote by \(F_{j,I}\) the fiber of \(P_{I}\) through the point \(q_{j}\) for any \(j\). For a given vector \((d_{1}, \dots, d_{n}) \in \mathbb{Z}_{\geq 0}^{n}\) we denote by NEWLINE\[NEWLINEs_{I}:= \sum_{i \in I}d_{i} \text{ and } S_{I}:=1+ |I| +s_{I}NEWLINE\]NEWLINE with \(I \subset \{1, \dots,n\}\). By the assumption that points are very general one has \(F_{i,I}\cap F_{j,I} = \emptyset\) for \(i\neq j\). Then \textit{the fiber dimension} of \(\mathcal{L}\) is defined as NEWLINE\[NEWLINE \text{fdim}(\mathcal{L}) := \prod_{i=1}^{n}(d_{i}+1) - \sum_{1\leq j \leq r; \, I\subset \{1, \dots, n\}; \, S_{I} \leq m_{j}} (-1)^{|I|}{ m_{j}-S_{I}+n \choose n} -1.NEWLINE\]NEWLINE The fiber expected dimension is defined as \(\text{efdim}(\mathcal{L}) = \max \{\text{fdim}( \mathcal{L} ), -1\}\) and we say that \(\mathcal{L}\) is \textit{fiber special} if \(\dim(\mathcal{L}) > \text{efdim}(\mathcal{L})\). The first main result of the note is the following.NEWLINENEWLINETheorem 1. For any linear system \(\mathcal{L}\) we have the following inequalities NEWLINE\[NEWLINE\dim(\mathcal{L}) \geq \text{efdim}(\mathcal{L}) \geq \text{edim}(\mathcal{L}).NEWLINE\]NEWLINE Another result gives us the following characterization of fiber non-special systems.NEWLINENEWLINE\textit{Theorem 2.} A linear system through two points in \((\mathbb{P}^{1})^{n}\) is fiber non-special.NEWLINENEWLINEAlso if there are more than two points, then then there are examples of fiber special systems (see Example 5.2).