In the paper under review the authors study surfaces in \(\mathbb{P}_3\) with triple points. A triple point can be written locally as \(x^3+y^3+z^3+\lambda xyz=0\) and is a non-simple surface singularity. For these singularities the invariants of the surface and its type in the classification of surfaces may change. Let \(\pi:\tilde{X}\longrightarrow X\) be the blow up of \(X\) at the triple points (otherwise \(X\) is smooth), the exceptional divisors are elliptic curves of self intersection \(-3\). The authors compute the invariants of the surface \(\tilde{X}\): \(c_1^2,~c_2,~\chi,~p_g,~q,~b_2,~h^{1,1}\) and show that: \[ c_1(\tilde{X})^2=K_{X_S}^2-3\nu,~~~~~~~~~~~~~~~~~~~c_2(\tilde{X})=c_2(X_S) -9\nu \] where \(c_i\) is the \(i\)-th Chern class, \(X_S\) is the smooth surface corresponding to \(X\) and \(\nu\) is the number of triple points. Moreover they discuss many bounds (Polar bounds, Miyaoka bounds, Spectrum bounds) finding the following upper bound for \(\nu\) depending on the degree \(d\) of the surface: \[ \begin{matrix} d&3&4&5&6&7&8&9&10&11&12\\ \nu\leq&1&1&5&10&17&29&42&60&81&107\end{matrix} \]
Then they classify all the surfaces with triple points up to the degree six. For degree four or less the classification is trivial, for degree five it is elementary, in this case they discuss many examples with some nice geometry. Then in degree six it becomes more complicated and the analysis of the exceptional curves of the first kind on \(\tilde{X}\) is fundamental for the classification. Finally the authors give an example of a surface of degree seven with 16 triple points. To construct it they consider special \(S_4\)-invariant polynomials of degree seven and some special orbit of points.