Many important moduli spaces arise as "tropical compactifications" of very affine varieties X introduced by Tevelev. For example, certain Chow and Hilbert quotients of Grassmannians including the moduli space $overline{M}_{0,n}$, of stable n-pointed rational curves, and moduli of del Pezzo surfaces. The Nef cone is a fundamental invariant of a proper variety X. Given a tropical compactification, its nef cone is sandwiched between two polyhedral cones. In this talk I'll define these cones and show that in the case $overline{M}_{0,n}$, the cones coincide for small n.

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