Very ampleness of \(d\)-standard classes on rational surfaces (Q1884781)

Let \(X_r\) be the surface obtained by blowing-up the complex projective plane \(\mathbb P^2\) along a curvilinear \(0\)-dimensional subscheme of length \(r\) of a reduced irreducible curve of degree \(d\). Inspired by work of Gimigliano and of Harbourne, the author gives sufficient conditions for the regularity, the spannedness, and the very ampleness of the so-called \(d\)-standard classes on \(X_r\), for \(d \geq 4\).