On freeness of divisors on $\mathbb P^2$

Abstract

Let $I\subset \mathbb C[x,y,z]$ be an ideal of height 2 and minimally generated by three homogeneous polynomials of the same degree. If $I$ is a locally complete intersection we give a criterion for $\mathbb C[x,y,z]/I$ to be arithmetically Cohen-Macaulay. Since the setup above is most commonly used when $I=J_F$ is the Jacobian ideal of the defining polynomial of a "quasihomogeneous" reduced curve $Y=V(F)$ in $\mathbb P^2$, our main result becomes a criterion for freeness of such divisors. As an application we give an upper bound for the degree of the reduced Jacobian scheme when $Y$ is a free rank 3 central essential arrangement, as well as we investigate the connections between the first syzygies on $J_F$, and the generators of $\sqrt{J_F}$.