The freeness conjecture for the Flenner–Zaidenberg second series

Let $C_k$ be the curve in the family ${\mathcal F}{\mathcal Z}_2(k)$, and let $mdr(f)$ denote the minimal degree of a Jacobian syzygy of its defining equation $f$. Let $\tau(C_k)$ be its total Tjurina number.

Freeness conjecture for ${\mathcal F}{\mathcal Z}_2$. The curves $C_k\in {\mathcal F}{\mathcal Z}_2(k)$ are free divisors and

$$mdr(f)=k-1$$

for all $k\geq 4$. In particular,

$$\tau(C_k)=3(k-1)^2$$

for every $k\geq 4$.

The conjecture was verified by direct computation for $4\leq k\leq 7$ using the parametrizations given in the paper; the general case remains open in the supplied text.