Osculatory behavior and second dual varieties of Del Pezzo surfaces

Abstract

Let S be a smooth surface embedded in a projective space, whose general osculating space has the expected dimension. Inside the dual variety of S one can consider the second discriminant locus, which parametrizes the hyperplane sections of S having some singular point of multiplicity ≥3. In this paper the various components of the second discriminant loci of Del Pezzo surfaces are investigated from a unifying point of view. This allows us to describe the second dual varieties of such surfaces and to understand their singular loci.