An asymptotic existence theorem for plane curves with prescribed singularities (Q2710517)

The author proves the following result: Let \((P_1, \dots, P_r)\) be a general \(r\)-tuple in \((\mathbb{P}^2)^r\) and let \(E\) be the linear system of plane curves of degree \(d\) passing through the point \(P_i\) \((i=1, \dots,r)\) with multiplicity at least \(m_i\). Then, given a positive integer \(m\), there exists an integer \(d'(m)\) such that, if \(m_i\leq m\) \((1\leq i\leq r)\) and \(d\geq d'(m)\), the system \(E\) has the expected dimension, i.e. \(e=\max(-1;d (d+ 3)/2 -\sum m_i(m_i+1)/2)\). Moreover, if \(e\geq 0\), then a general curve in \(E\) is irreducible, smooth away from the \(P_i\) and has an ordinary singularity of multiplicity \(m_i\) at each \(P_i\). The techniques of the paper are inspired by a method introduced by \textit{J. Alexander} and \textit{A. Hirschowitz} [Invent. Math. 140, No. 2, 303-325 (2000; Zbl 0973.14026)], but with a different approach that yields conclusions on the irreducibility and smoothness. It is interesting to observe that the result above holds also when the expected dimension is small, so that it is new with respect to previous results [see for instance \textit{G.-M. Greuel}, \textit{C. Losse} and \textit{E. Shustin}, Trans. Am. Math. Soc. 350, No. 1, 251-274 (1998; Zbl 0889.14010)]. As far as the number \(d'(m)\) is concerned, an explicit, doubly exponentially growing, value is computed.