zero.

[[@MMN Proposition 6.1]]{} \[pro:amaci-props\] Let $I = I_{a,b,c,\alpha,\beta,\gamma}$ be defined as above. Assume, without loss of generality, that $0 \leq \alpha \leq \beta \leq \gamma$.

1.  If $\alpha = 0$, then $R/I$ has socle type $2$ with socle degrees $a+\beta+c-3$ and $a+b+\gamma -3$; thus $R/I$ is level if and only if $b - \beta = c - \gamma$.

2.  If $\alpha > 0$, then $R/I$ has socle type $3$ with socle degrees $\alpha + b + c -3$, $a+\beta+c-3$, and $a+b+\gamma -3$; thus $R/I$ is level if and only if $a - \alpha = b - \beta = c - \gamma$.

3.  Moreover, the minimal free resolution of $R/I$ has the form $$\label{eqn:resolution}
                    0
                    \rightarrow
                    \begin{array}{c} R(-a-b-\gamma) \\ \oplus \\ R(-a-\beta-c) \\ \oplus \\ R^n(-\alpha-b-c) \end{array}
                    \rightarrow
                    \begin{array}{c} R(-a-\beta-\gamma) \\ \oplus \\ R(-\alpha-b-\gamma) \\ \oplus \\  R(-\alpha-\beta-c) \\ \oplus \\
                                     R(-a-b) \\ \oplus \\ R(-a-c) \\ \oplus \\ R^n(-b-c) \end{array}
                    \rightarrow
                    \begin{array}{c} R(-\alpha-\beta-\gamma) \\ \oplus \\ R(-a) \\ \oplus \\ R(-b) \\ \oplus \\ R(-c) \end{array}
                    \rightarrow
                    R
                    \rightarrow
                    R/I
                    \rightarrow
                    0$$ where $n = 1$ if $\alpha > 0$ and $n = 0$ if $\alpha = 0$.

Moreover, we see that in characteristic zero the weak Lefschetz property 