Quintics with three triple points, sextics with five and degenerations (Q2487838)

Let \(X \subset {\mathbb P^3}\) be a quintic surface with three ordinary triple points \(P_1, P_2,P_3\) and let \(S\) be the minimal resolution of the singularities of \(X\). Using adjunction, one checks that if the \(P_i\) are in general position then \(S\) is a \(K3\) surface, while if the \(P_i\) are collinear then the geometric genus of \(S\) is equal to 2. This remark shows that if a quintic \(X\) with three collinear triple points is the limit of a family of quintics with three non collinear triple points, then \(X\) is not general among the quintics with three collinear triple points. The author works out precisely the extra conditions on the singularities that characterize these ``limit quintics''. A similar analysis is carried out also for sextics with five coplanar triple points, that are limits of sextics with five non coplanar triple points.