Blowup and specialization methods for the study of linear systems (Q2870821)

The paper consists of notes about blowup-based tools for linear system.NEWLINENEWLINELet \(p_1, \dots, p_r\) be general points in \(\mathbb{P}^2\) and let \(m_1, \dots, m_r\) be positive integers. By \(\mathcal{L}(d;m_1, \dots, m_r)\) we denote the system of plane curves of degree \(d\) with multiplicity at least \(m_j\) at \(p_j\), \(j=1, \dots, r\). \(\mathcal{L}(d;m_1^{\times s_1}, \dots, m_r^{\times s_r})\). is a system with repeated multiplicities. The expected dimension of \(\mathcal{L}(d;m_1, \dots, m_r)\) is NEWLINE\[NEWLINE\mathrm{edim}(\mathcal{L}(d;m_1, \dots, m_r)=\max \{\mathrm{vdim}(\mathcal{L}(d;m_1, \dots, m_r)), -1 \}, NEWLINE\]NEWLINE where the virtual dimension, \(\mathrm{vdim}(\mathcal{L}(d;m_1, \dots, m_r))\), is \(\frac{d(d+3)}{2}-\sum_{j=1}^r {m_j+1 \choose 2}\). A system is special if its effective dimension is strictly greater than the expected one.NEWLINENEWLINERecent years have seen significant advances in the understanding of linear systems with imposed multiple points. The case of points in general position is an important case, with several relevant contributions to the open conjectures of Nagata-Biran-Szemberg and Segre-Harbourne-Gimigliano-Hirschowitz. Most of these rely to some extent on semicontinuity and degeneration methods, which often allow setting up induction arguments on the multiplicity or the number of points.NEWLINENEWLINEThe formalism of blowups has become an essential tool in te study of linear systems with multiple points, especially when using degeneration methods: the geometry of the variety blown up at the imposed points is important; induction arguments often lead to consider points that are not in general position, but ``infinitely near'', i.e. on blowups; useful degenerations are often built by blowing up the total space of some family.NEWLINENEWLINEThe author aims to overview the set of blowup-based tools that are being used for specializing and degenerating linear systems, with emphasis on clusters of infinitely near points and Ciliberto-Miranda's \textit{blowup and twist}.NEWLINENEWLINEThe author takes a rather elementary approach, which should serve as a friendly introduction and guide to original research articles. Sometimes full proof are not given or the exposition restricts to particular cases for the sake of simplicity; in such case the author includes references to the existing bibliography. In particular, the author deals only with linear systems of curves on smooth surfaces defined over the field of complex numbers.