Gröbner bases for spaces of quadrics of codimension 3

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