Rational curves on primitive symplectic varieties of OG 6s -type

We prove that any ample class on a primitive symplectic variety that is a locally trivial deformation of O'Grady's singular 6-dimensional example is proportional to the first Chern class of a uniruled divisor. This result answers a question of Lehn-Mongardi-Pacienza (Lehn et al. 2021, Remark 4.7) extending their result (Lehn et al. 2021, Theorem 1.3) for primitive symplectic varieties of this deformation type. In order to get our result we produce examples of positive uniruled divisors on singular moduli spaces of sheaves that are locally trivial deformations of O'Grady's example. We show that all possible square and divisibility of polarizations arise on such moduli spaces, hence we conclude by maximality of the monodromy group of this deformation class of singular symplectic varieties. Finally we provide some considerations on the smooth case, motivating why our techniques fail on the smooth setting and pointing out what information is needed to conclude the existence of positive uniruled divisors on smooth O'Grady's sixfolds starting from our result.