:=& s+2-c, \mbox{ and}\\
            M :=& s + 2 - (\alpha + \beta + \gamma).
        \end{split}$$ Then $I$ has a semistable syzygy bundle if and only if the following conditions all hold:

1.  $0 \leq M$,

2.  $0 \leq A \leq \beta  + \gamma$,

3.  $0 \leq B \leq \alpha + \gamma$, and

4.  $0 \leq C \leq \alpha + \beta$.

Using [@Br Corollary 7.3] we have that $I$ has a semistable syzygy bundle if and only if

1.  $\max\{a, b, c, \alpha + \beta + \gamma\} \leq s+2$,

2.  $\min\{\alpha + \beta + c, \alpha + b + \gamma, a + \beta + \gamma\} \geq s+2$, and

3.  $\min\{a+b, a+c, b+c\} \geq s+2$.

Notice that condition (a) is equivalent to $A, B, C,$ and $M$ being non-negative. Moreover, condition (b) is equivalent to the upper bounds on $A, B,$ and $C$. We claim that condition (c) follows directly from condition (a).

Indeed, by condition (a) we have that $C+M \geq 0$ and so $A+B+C+M = s+2 \geq A+B = 2(s+2) - a - b$, thus $a + b \geq s+2$. Similarly, we have $a + c \geq s+2$ and $b+c \geq s+2$. Thus condition (c) holds if condition (a) 