On the equations defining minimal varieties

The classification of all projective varieties of minimal degree is due to the successive contributions of various authors, and spreads over a century. More than hundred years ago, in 1886, Del Pezzo solved the surface case, then, in 1907, Bertini extended the characterization to irreducible varieties of any dimension. He found a class of rationally ruled varieties, for which Harris in 1976 gave a nice algebraic description: he proved that they are a particular class of determinantal varieties, which he called scrolls. Finally, in 1981, Xambo completed the classification for varieties with more than one irreducible component. This class of varieties includes the blow-ups of all monomial varieties of codimension 2. De Concini-Eisenbud-Procesi state that "the precise equations satisfied by reducible subvarieties of minimal degree remain mysterious", hence they "stop short of giving a normal form for the equations of each type" of them. This problem is solved in the present paper.