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We study rank-two reflexive sheaves on $\mathbb{P}^3$ with $c_2 =4$,expanding on previous results for $c_2\le3$. We show that every spectrum notpreviously ruled out is realized. Moreover, moduli spaces are studied anddescribed in detail for $c_1=-1$ or $0$ and $c_3\ge8$.

Rank-two reflexive sheaves on the projective space with second Chern class equal to four