Polarized 3-folds in a codimension 10 weighted homogeneous $F_4$ variety

TL;DR

This paper constructs a specific weighted homogeneous variety related to the exceptional Lie group F4, computes its graded ring and Hilbert series, and develops families of polarized 3-folds as weighted complete intersections.

Contribution

It provides an explicit construction and algebraic description of a codimension 10 weighted F4 variety and introduces new families of polarized 3-folds derived from it.

Findings

Explicit construction of $w ext{}\Sigma F_4(\mu,u)$

Formula for the Hilbert series in representation-theoretic terms

Families of polarized 3-folds as weighted complete intersections

Abstract

We give the construction of a codimension 10 weighted homogeneous variety $w\Sigma F_4(\mu,u)$ corresponding to the exceptional Lie group $F_4$ by explicit computation of its graded ring structure. We give a formula for the Hilbert series of the generic weighted $w\Sigma F_4(\mu,u)$ in terms of representation theoretic data of $F_4$. We also construct some families of polarized 3-folds in codimension 10 whose general member is the weighted complete intersection of some $w\Sigma F_4(\mu,u)$.