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In terms of the gauged nonlinear $\sigma$-models, we describe some resultsand implications of solving the following problem: Given a smooth symplecticmanifold as target space with a quasi-free Hamiltonian group action, performthe symplectic blowing up of the point singularity and identify the blow-upmodes in the corresponding (gauged) $\sigma$-model. Both classical and quantumaspects of the construction are explained, along with illustrating examplesfrom the toric projective space and the K\"ahler manifold. We also discussrelated problems such as the origin of Mirror symmetry and the quantumcohomologies.(Talk to be given at ICHEP94, Glasgow, July 20-27.)Comment: 12 pages, LaTex, THEP 94/17 (5 figures not included

Some features of blown-up non-linear -models