Claudio Fontanari Title: A fundamental theorem of the geometry on algebraic surfaces and the splitting principle Abstract: My talk finds its roots in an old paper by Beniamino Segre carrying the same title and going back to 1938. I am going to frame it into its historical and mathematical context and to present in modern terms a splitting principle stated there with an incomplete proof. This is joint work with Camilla Felisetti (https://arxiv.org/abs/2105.00892).