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Unexpected surfaces singular on lines in \(\mathbb{P}^3\). (English) Zbl 1467.14018

In the paper under review, the authors study special linear series of surfaces in \(\mathbb{P}^{3}\) that are singular along some general lines, i.e., those non-empty series where the conditions imposed by the multiple lines are not independent. The main result of the paper provides us four surfaces arising from projective linear series with a single reduced member. Let us denote by \(\mathcal{L} = \mathcal{L}_{d}(m_{1}, \dots m_{s})\) the linear series of surfaces of degree \(d\) in \(\mathbb{P}^{3}\) passing through \(s\) general lines with assigned multiplicities \(m_{1}, \dots, m_{s}\). If \(d < m_{i}\) for some \(i\), then clearly the series \(\mathcal{L}\) is empty, so one assumes that \(d \geq \max\{m_{1}, \dots, m_{s}\}\). Moreover, we denote by \(\mathcal{L}_{d}(m^{\times s})\) a linear series of surfaces of degree \(d\) with \(s\) lines of the same multiplicity \(m\). The main result of the paper can be formulated as follows.
Main Result. The following system are special of (affine) dimension 1:
(A) \(\mathcal{L}_{10}(3^{\times 4}, 1^{\times 5})\);
(B) \(\mathcal{L}_{12}(4,3^{\times 5})\);
(C) \(\mathcal{L}_{12}(3^{\times 6}, 2)\);
(D) \(\mathcal{L}_{20}(6^{\times 5},1)\).
Thus there is a single surface of the given degree vanishing to given order along the given number of general lines.

MSC:

14C20 Divisors, linear systems, invertible sheaves
14E05 Rational and birational maps

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