The Segre and Harbourne-Hirschowitz conjectures (Q2754413)

For every \(P\in\mathbb{P}^2\), let \(mP\) the subscheme of \(\mathbb{P}^2\) with \(({\mathcal I}_P)^m\) as ideal sheaf (a fat point of order \(m)\). Hence \(mP\) has length \(m(m+1)/2\). Fix integers \(m_i>0\), \(1\leq i\leq s\), and let \(Z\subseteq \mathbb{P}^2\) be the union of \(s\) general fat points of order \(m_1,\dots, m_s\). It was conjectured that, with a few exceptions, \(Z\) has maximal rank, i.e. that for all integers \(t>0\) the restriction map \(H^0(\mathbb{P}^2, {\mathcal O}_{\mathbb{P}^2} (t)) \to H^0(Z,{\mathcal O}_Z(t))\) is injective or surjective. \textit{B. Harbourne} [in: Algebraic geometry, Proc. Conf., Vancouver 1984, CMS Conf. Proc. 6, 95-111 (1986; Zbl 0611.14002)] and \textit{A. Hirschowitz} [J. Reine Angew. Math. 397, 208-213 (1989; Zbl 0686.14013)] conjectured this result and made a very precise description of the exception cases.NEWLINENEWLINENEWLINESeveral papers by several authors are devoted to the proof of particular cases of this conjecture. In the paper under review, the authors prove that this conjecture is equivalent to an old conjecture of B. Segre and deduce, under the assumption of the truth of this conjecture, some general results about the structure of the linear systems of all degree \(t\) plane curves with multiplicity \(m_1,\dots, m_s\) at general points \(P_1,\dots, P_s\).NEWLINENEWLINEFor the entire collection see [Zbl 0971.00013].