Constructing projective varieties in weighted flag varieties

TL;DR

This paper develops methods to compute Hilbert series and defining equations of weighted flag varieties, enabling the construction of new polarized projective varieties in specific codimensions using computational techniques.

Contribution

It introduces a computer-aided approach to determine equations of weighted flag varieties, expanding the toolkit for constructing complex algebraic varieties.

Findings

Computed Hilbert series for general weighted flag varieties

Developed a method to find defining equations computationally

Constructed new families of polarized projective varieties in codimensions 8 and 6

Abstract

We compute the Hilbert series of general weighted flag varieties and discuss a computer-aided method to determine their defining equations. We apply our results to weighted flag varieties coming from the Lie groups of type G_2 and GL(6), to construct some families of polarised projective varieties in codimensions 8 and 6, respectively.