Gröbner bases for spaces of quadrics of codimension 3

Let R=⊕_i≥0R_i be an Artinian standard graded K-algebra defined by quadrics. Assume that dimR₂≤3 and that K is algebraically closed, of characteristic ≠2. We show that R is defined by a Gröbner basis of quadrics with, essentially, one exception. The exception is given by Kx,y,z]/I where I is a complete intersection of three quadrics not containing a square of a linear form.