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We investigate the universal Severi variety of rational curves on K3 surfaces, which parametrizes irreducible rational curves in a fixed class on varying K3 surfaces of fixed genus. We investigate the conjectured irreducibility of this space in the case of primitive classes using the deformation theory of stable maps and obtain partial results. Furthermore, we give a short proof of the irreducibility of universal Severi varieties of high‐genus curves on K3 surfaces.

The universal Severi variety of rational curves on K3 surfaces