Gröbner Bases for Spaces of Quadrics of Low Codimension

Abstract

Let R=⊕i≥0Ri be a quadratic standard graded K-algebra. Backelin has shown that R is Koszul provided dimR2≤2. One may wonder whether, under the same assumption, R is defined by a Gröbner basis of quadrics. In other words, one may ask whether an ideal I in a polynomial ring S generated by a space of quadrics of codimension ≤2 always has a Gröbner basis of quadrics. We will prove that this is indeed the case with, essentially, one exception given by the ideal I=(x2, xy, y2−xz, yz)⊂K[x, y, z]. We show also that if R is a generic quadratic algebra with dim R2<dimR1 then R is defined by a Gröbner basis of quadrics.