For coprime rank r and first Chern class aH, the 2N-th Betti number of the moduli space of sheaves on P^2 stabilizes once c2 >= N + floor((r-1)/(2r)a^2 + (r^2+1)/2).
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On the stabilization of the Betti numbers of the moduli space of sheaves on $\mathbb{P}^2$
For coprime rank r and first Chern class aH, the 2N-th Betti number of the moduli space of sheaves on P^2 stabilizes once c2 >= N + floor((r-1)/(2r)a^2 + (r^2+1)/2).