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We study birational geometry of the moduli space of parabolic bundles over a projective line, in the framework of Mori's program. We show that the moduli space is a Mori dream space. As a consequence, we obtain the finite generation of the algebra of type A conformal blocks. Furthermore, we compute the H-representation of the effective cone which was previously obtained by Belkale. For each big divisor, the associated birational model is described in terms of moduli space of parabolic bundles.

Finite Generation of the Algebra of Type A Conformal Blocks via Birational Geometry