Dessins d'Enfants and Hypersurfaces with Many $A_j$-Singularities

We show the existence of surfaces of degree $d$ in ${\text{dP}}^3(\text{dC})$ with approximately $\frac{3j+2}{6j(j+1)}{d}^3$ singularities of type $A_j, 2\le j\le d-1$. The result is based on Chmutov’s construction of nodal surfaces. For the proof we use plane trees related to the theory of Dessins d’Enfants. Our examples improve the previously known lower bounds for the maximum number ${\mu }_{A_j}(d)$ of $A_j$-singularities on a surface of degree $d$ in most cases. We also give a generalization to higher dimensions which leads to new lower bounds even in the case of nodal hypersurfaces in ${\text{dP}}^n,n\ge 5$. To conclude, we work out in detail a classical idea of B. Segre which leads to some interesting examples, e.g. to a sextic with 36 cusps.