Base point free pencils of small degree on certain smooth curves (Q1973439)

The author gives two sufficient conditions for every base point free \(g^1_m\) of a plane curve of degree \(\geq m\) to be induced by a pencil of lines or conics in \({\mathbb{P}}^2\) in terms of the degree and its singularities. More precisely, given a plane curve \(C\) of degree \(d\) and a base point free pencil of degree \(m \leq d\) on the normalization \(\hat{C}\), an irreducible curve in \({\mathbb{P}}^1 \times {\mathbb{P}}^2\) is constructed that represents a special cohomology class. Then, the author uses Castelnuovo inequalities and singularity conditions to show that this curve lies on an irreducible surface \(S\subset {\mathbb{P}}^1 \times {\mathbb{P}}^2\) of degree 4 or 5. Since those \(S\) can be easily classified, it can be shown that the only possible cases for such \(g^1_m\) are induced by a pencil of lines or a pencil of conics in \({\mathbb{P}}^2\). For the second bound, an example is given in which the inequality is sharp.