Quadratic Ideals in Six Variables and the Eisenbud--Green--Harris Conjecture

In this paper, we study the Eisenbud–Green–Harris (EGH) conjecture for ideals generated by quadrics. We establish a sharp lower bound for the dimension of the cubic component of an ideal generated by a regular sequence of six quadrics and two additional quadrics in six variables. Furthermore, we prove the Eisenbud–Green–Harris conjecture for almost complete intersections of quadrics in six variables.