Numerical conditions for the boundedness of foliated surfaces

We show that the set of Hilbert functions $P(m)=\chi(mK_\mathcal{F})$ of 2-dimensional foliated canonical models with fixed $K_\mathcal{F}^2$, $K_\mathcal{F} \cdot K_X$ and $i_\mathbb{Q}(\mathcal{F})$ is finite. As a consequence, we deduce that two results on the effective birationality and boundedness of foliated canonical models with fixed Hilbert function still hold when only $K_\mathcal{F}^2$, $K_\mathcal{F} \cdot K_X$ and $i_\mathbb{Q}(\mathcal{F})$ are fixed. We then give examples further investigating the properties of families of canonical models, and study particular cases in which some of the conditions are not necessary.