On the curves through a general point of a smooth surface in \(\mathbb{P}^ 3\) (Q1895772)

In this paper the authors study the geometry of correspondences with null trace of low degree, on a smooth surface in \(\mathbb{P}^3\) of degree \(d\geq 5\). In their main theorem they prove that given a smooth surface in \(\mathbb{P}^3\) of degree \(d\geq 5\) and a correspondence of degree \(n\) with null trace, then \(n\geq d-2\). Furthermore, they show for which correspondences the above inequality is an equality. They use their main theorem to get several applications on linear series on families of curves on a given surface.