Diptych varieties. II: Polar varieties

TL;DR

This paper extends the construction of diptych varieties to cases where the product of parameters is less than or equal to four, and introduces new quasihomogeneous spaces related to algebra of polars.

Contribution

It completes the existence proof of diptych varieties for all parameter cases and constructs new classes of quasihomogeneous spaces based on algebra of polars.

Findings

Proved existence of diptych varieties for all cases with de ≤ 4.

Constructed new quasihomogeneous spaces for groups like GL(2)×G_m^r.

Extended the algebraic framework of polar varieties.

Abstract

Part I introduced diptych varieties $V_{ABLM}$ and gave a rigorous construction of them in the case $d,e\ge 2$ and $de>4$. Here we prove the existence of $V_{ABLM}$ in all the cases with $de\le4$. At the same time we construct some classes of interesting quasihomogeneous spaces for groups such as $\GL(2)\times\G_m^r$ based on the algebra of polars.