A blowup algebra for hyperplane arrangements

Abstract

It is shown that the Orlik–Terao algebra is graded isomorphic to the special fiber of the ideal $I$ generated by the $\left(n-1\right)$-fold products of the members of a central arrangement of size $n$. This momentum is carried over to the Rees algebra (blowup) of $I$ and it is shown that this algebra is of fiber-type and Cohen–Macaulay. It follows by a result of Simis and Vasconcelos that the special fiber of $I$ is Cohen–Macaulay, thus giving another proof of a result of Proudfoot and Speyer about the Cohen–Macaulayness of the Orlik–Terao algebra.