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In this article, we classify the irreducible components of moduli stacks oftorsion free sheaves of rank 2 on K3 surfaces of Picard number 1. For ruledsurfaces, the components of moduli stacks of torsion free sheaves wereclassified by C.Walter. Moreover, by virtue of our result, we classify theirreducible components of Brill-Noether loci of Hilbert schemes of points on K3surfaces.

Classifying the irreducible components of moduli stacks of torsion free sheaves on K3 surfaces and an application to Brill-Noether theory