Hodge theory and syzygies of the Jacobian ideal

Let $f$ be a homogeneous polynomial, defining a principal Zariski open set $D(f)$ in some complex projective space $\mathbb{P}^n$ and a Milnor fiber $F(f)$ in the affine space $\mathbb{C}^{n+1}$. Let$f_0, . . . , f_n$ denote the partial derivatives of $f$ with respect to $x_0, . . . , x_n$ and consider syzygies $a_0f_0 + a_1f1 + a_nf_n = 0$, where $a_j$ are homogeneous polynomials of the same degree $k$. Using the mixed Hodge structure on $D(f)$ and $F(f)$, one can obtain information on the possible values of $k$.