Calculating Generators of Multigraded Algebras

TL;DR

This paper introduces algorithms to compute generators of multigraded algebras associated with p-divisors, including cases with torus actions, demonstrated through the Cox ring of a del Pezzo surface.

Contribution

It presents new algorithms for finding generators of multigraded algebras, extending to cases with torus actions, and applies them to a specific geometric example.

Findings

Successfully computed generators for the Cox ring of a degree 5 del Pezzo surface.

Provided algorithms applicable to arbitrary p-divisors on varieties.

Extended the algorithm to cases where the variety admits a torus action.

Abstract

We present an algorithm to find generators of the multigraded algebra A associated to an arbitrary p-divisor D on some variety Y. A modified algorithm is also presented for the case where Y admits a torus action. We demonstrate our algorithm by computing generators for the Cox ring of the smooth del Pezzo surface of degree 5.