  We construct new "virtually smooth" modular compactifications of spaces of
maps from nonsingular curves to smooth projective toric varieties. They
generalize Givental's compactifications, when the complex structure of the
curve is allowed to vary and markings are included, and are the toric
counterpart of the moduli spaces of stable quotients introduced by Marian,
Oprea, and Pandharipande to compactify spaces of maps to Grassmannians. A brief
discussion of the resulting invariants and their (conjectural) relation with
Gromov-Witten theory is also included.

Algebraic Geometry