Toric divisors in depth

In probably the final instalment of me filling in my understanding of toric varieties for a while, I cover some of the approaches to studying divisors on toric varieties. A key approximation result is the ability to find a torus-invariant divisor inside any rational equivalence class. This immediately transports essentially any interesting question about divisors to the arena of combinatorics.

This approach is effective for testing whether or not various adjectives can be attached to a given divisor. To a torus-invariant divisor D one can associate a piecewise-linear function \varphi_D that contains enough information to recover the divisor, but is often more amenable to study. The main entries in the dictionary between divisors and piecewise-linear functions are…

\text{basepoint-free}\longleftrightarrow\text{convex}

\text{ample}\longleftrightarrow\text{strictly convex}

It actually follows from the first correspondence that basepoint-free divisors are exactly the nef divisors on a toric orbifold, by observing that they are also the divisors whose associated piecewise-linear functions are convex.

Returning to Mori theory gives, in my opinion, the best way to view to these correspondences. The functions \varphi_D can be used to calculate intersection pairings between torus-invariant divisors and curves and demanding convexity picks out the nef cone, while strict convexity restricts to the interior of the nef cone, which consists of the ample divisors.

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