Betti numbers of GIT quotients of products of projective planes

TL;DR

This paper investigates the topology of GIT quotients of multiple projective planes under SL_3(C) action, providing a method to compute their intersection Poincaré polynomials and explicit Betti numbers for specific cases.

Contribution

It introduces a strategy to compute intersection Poincaré polynomials of GIT quotients of products of projective planes, with explicit formulas for Betti numbers when n=6.

Findings

Developed a general method for intersection Poincaré polynomial calculation.

Derived explicit Betti number formulas for the case n=6.

Connected Betti number computations to combinatorics of weights.

Abstract

We study the GIT quotients for the diagonal action of the algebraic group $SL_3(\mathbb{C})$ on the $n$-fold product of $\mathbb{P}^2(\mathbb{C})$: in particular we determine a strategy in order to determine the (intersection) Poincar\'{e} polynomial of any quotient variety. In the special case $n=6$ we determine an explicit formula for the (intersection) Betti numbers of a quotient variety, depending only on the combinatorics of the weights of the polarization $m\in \mathbb{Z}^6_{>0}$.