ToricHigherDirectImages -- computations involving pushforwards and higher direct images of toric maps

Given a morphism of varieties $\varphi \colon X \rightarrow Y$, we have the pushforward functor $\varphi_*$ from the category of coherent sheaves on $X$ to coherent sheaves on $Y$. This functor is not right-exact, and so there is a right-derived functor $R^i \varphi_*$ from the bounded derived category $D^b(X)$ to $D^b(Y)$, which we call the higher direct image functor. When both $X$ and $Y$ are toric varieties, coherent sheaves arise from finitely generated multigraded modules over their Cox rings. Thus, for a given coherent sheaf $\mathcal{F}$, there are finitely generated graded modules over the Cox ring of $Y$ which sheafifies to $R^i \varphi_* \mathcal{F}$.

The purpose of this package is to compute (higher) direct images of toric morphisms $\varphi \colon X \rightarrow Y$ between smooth projective toric varieties. This is currently implemented in two situations:

1) When $\varphi$ is a toric fibration, i.e. a surjective toric morphism with $\varphi_* \mathcal{O}_X = \mathcal{O}_Y$, then the method \texttt{HDI} allows one to compute the higher direct images of a line bundle $\mathcal{O}_X(D)$ on $X$. In "Reduced \v{C}ech complexes and computing higher direct images under toric maps", M. Roth provides a constructive method for producing this module and S. Zotine adapts that method into an algorithm. This package provides the implementation of this algorithm.

For instance, if $X$ is a blowup of $\mathbb{P}^2$, then the first higher direct image of $O_X(3E)$ is nontrivial, where $E$ is the exceptional divisor.

2) When $\varphi \colon Y \rightarrow Y$ is a toric Frobenius map, i.e. induced from the natural morphism on the dense torus given by raising all of the coordinates to the same power, then the method \texttt{frobeniusDirectImage} allows one to compute pushforwards of any coherent sheaf. For instance, the pushforward of the trivial bundle on $\mathbb{P}^2$ by the second Frobenius map splits into 4 line bundles.

References

M. Roth and S. Zotine, \textit{Reduced \v{C}ech complexes and computing higher direct images under toric maps}, to appear on arXiv

Contributors

The following people have generously contributed code, improved existing code, or enhanced the documentation: Mahrud Sayrafi, and Gregory G. Smith.

In case 1), we have only implemented the computation for line bundles, and if the target is simplicial. The only finite maps for which the pushforward is implemented are the Frobenius maps.