Monoidal and submonoidal surfaces, and Cremona transformations

Abstract

We study irreducible surfaces of degree d in $\mathbb{P}^3$ that contain a line of multiplicity d-1 (monoidal surfaces) or d-2 (submonoidal surfaces). We relate them to congruences of lines and Cremona transformations. Many of our results are not new and can be found in the classical literature, we give them modern proofs. In the last section, we extend some of our results to hypersurfaces of arbitrary dimension. We define two commuting Cremona involutions in the ambient space associated to a linear subspace of multiplicity d-2 contained in the hypersurface. Both leave the hypersurface invariant, but one acts as the identity on the hypersurface.