The non-emptiness criterion for coherent systems on the projective line

Let $G(\alpha;n,d,k)$ denote the moduli space of $\alpha$-stable coherent systems of type $(n,d,k)$ on the projective line. Let $a\ge2$, $1\le t\le n-1$, $k\ge n$, and suppose that

$$k((a+1)n-t-k)\ge n^2-1.$$

Non-emptiness conjecture. The space $G(\alpha;n,an-t,k)$ is non-empty for some $\alpha>0$ if and only if one of the following holds:

• $at<k<an$ and $(a-1)t\le a(an-k)+(a-2)n$;
• $k\le at$.

This conjecture proposes a complete criterion under the necessary numerical condition from the preceding theorem. The cited results establish several non-existence cases and prove non-emptiness in corresponding boundary cases, but the equivalence stated here is not resolved in the supplied text.