We produce complete, non-compact, Riemannian metrics with positive constant \sigma_2-curvature on a sphere of dimension n>4, with a prescribed singular set given by a disjoint union of closed submanifolds whose dimension is positive and strictly less than (n−n−\sqrt 2)/2. The \sigma_2-curvature in conformal geometry is defined as the second elementary symmetric polynomial of the eigenvalues of the Schouten tensor, which yields a fully non-linear PDE for the conformal factor. We show that the classical gluing method of Mazzeo-Pacard (JDG 1996) for the scalar curvature still works in the fully non-linear setting. This is a consequence of the conformal properties of the equation, which imply that the linearized operator has good mapping properties in weighted spaces. This is joint work with María Fernanda Espinal. This seminar is part of the activities of the Dipartimento di Eccellenza CUP B83C23001390001 and it is funded by the European Union – Next Generation EU.