follows for certain choices of the parameters.

[[@MMN Theorem 6.2]]{} \[pro:amaci-not-3\] Let $K$ be an algebraically closed field of characteristic zero. Then $R/I_{a,b,c,\alpha,\beta,\gamma}$ has the weak Lefschetz property if $a + b + c + \alpha + \beta + \gamma \not\equiv 0 \pmod{3}$.

 

Semi-stability
--------------

 

The syzygy module $\operatorname{syz}{I}$ of $I = I_{a,b,c,\alpha,\beta,\gamma}$ fits into the exact sequence $$0
    \longrightarrow
        \operatorname{syz}{I}
    \longrightarrow 
        R(-\alpha-\beta-\gamma) \oplus R(-a) \oplus R(-b) \oplus R(-c)
    \longrightarrow 
        I_{a,b,c,\alpha,\beta,\gamma}
    \longrightarrow 
        0.$$ The sheafification $\widetilde{\operatorname{syz}{I}}$ is a rank 3 bundle on ${\mathbb{P}}^2$, and it is called the [*syzygy bundle*]{} of $I$. Recall that a vector bundle $E$ on projective space is said to be [*semistable*]{} if, for every coherent subsheaf $F \subset E$, the following inequality holds: $$\frac{c_1(F)}{rk(F)} \leq \frac{c_1(E)}{rk(E)}.$$

We analyse when $I_{a,b,c,\alpha,\beta,\gamma}$ has a semistable syzygy bundle. (Note, the slightly awkward definition of $s$ in the following is kept for consistency with [@MMN Section 7], the starting point of this work.)

\[pro:semistable\] Let $K$ be an algebraically closed field of characteristic zero. Further, let $I = I_{a,b,c,\alpha,\beta,\gamma}$, and define the following rational numbers $$\begin{split}
            s :=& \frac{1}{3}(a + b + c + \alpha + \beta + \gamma)-2, \\
            A :=& s+2-a, \\
            B :=& s+2-b, \\ 
            C 