The Koszul property for spaces of quadrics of codimension three

In this paper we prove that, if k is an algebraically closed field of characteristic different from 2, almost all quadratic standard graded k-algebras R such that dim_k(R_2)=3 are Koszul. More precisely, up to graded k-algebra homomorphisms and trivial fiber extensions, we find out that only two (or three, when the characteristic of k is 3) algebras of this kind are non-Koszul. Moreover, we show that there exist nontrivial quadratic standard graded k-algebras with dim_k(R_1)=4, dim_k(R_2)=3 that are Koszul but do not admit a Gröbner basis of quadrics even after a change of coordinates, thus settling in the negative a question asked by Conca.