  We refine the affine classification of real nets of quadrics in order to
obtain generic curvature loci of regular $3$-manifolds in $\mathbb{R}^6$ and
singular corank $1$ $3$-manifolds in $\mathbb{R}^5$. For this, we characterize
the type of the curvature locus by the number and type of solutions of a system
of equations given by 4 ternary cubics (which is a determinantal variety in
some cases). We also study how singularities of the curvature locus of a
regular 3-manifold can go to infinity when the manifold is projected
orthogonally in a tangent direction.
