Multigraded Apolarity

TL;DR

This paper extends apolarity methods to compute various tensor ranks on toric varieties, providing bounds and explicit calculations for specific surfaces and weighted projective planes.

Contribution

It introduces a generalized framework for rank computation on toric varieties and derives bounds on cactus rank, with explicit examples for several surfaces.

Findings

Upper bound on cactus rank for toric varieties

Explicit rank, border rank, and cactus rank computations for specific surfaces

Extension of apolarity methods to multigraded settings

Abstract

We generalize methods to compute various kinds of rank to the case of a toric variety $X$ embedded into projective space using a very ample line bundle $\mathcal{L}$. We find an upper bound on the cactus rank. We use this to compute rank, border rank, and cactus rank of monomials in $H^0(X,\mathcal{L})^*$ when $X$ is $\mathbb{P}^1 \times \mathbb{P}^1$, the Hirzebruch surface $\mathbb{F}_1$, the weighted projective plane $\mathbb{P}(1,1,4)$, or a fake weighted projective plane.