The Cayley-Bacharach property and the Levinson-Ullery conjecture

In this paper, we study the geometric configurations of a finite set of points having the Cayley-Bacharach property in the $n$-dimensional projective space $\bbP^n$. Our main contribution is the proof of the Levinson-Ullery conjecture for the previously unsolved case where $d=4$ and $r\ge 1$.