Divisoriality and 𝑚-canonical ideal for quadratic quotients of the Rees algebra

Let R be a one-dimensional Cohen-Macaulay local ring and I an ideal of R. It is well known that both the classical construction of Nagata’s idealization R ⋉ I and a recent construction known as amalgamated duplication, are Gorenstein when I is a canonical ideal of R. This property holds also for a more general family of rings, the quadratic quotients of the Rees algebra associated to R with respect to an ideal I and the elements a, b ∈ R, defined in an attempt to provide a unified approach of the two construction above. Since for a one-dimensional Noetherian domain the Gorenstein property is equivalent to the divisorial property, our pourpose is to understand, in a more general setting, when a quadratic quotient R(I)a,b is divisorial when I is an m-canonical ideal of R.