For any smooth projective toric variety, a contraction morphism from stable maps to quasimaps is constructed on a closed substack, and is proven surjective when the target is Fano.
Orbifold quasimap theory
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The contraction morphism between maps and quasimaps to toric varieties
For any smooth projective toric variety, a contraction morphism from stable maps to quasimaps is constructed on a closed substack, and is proven surjective when the target is Fano.