GIT Quotients of Products of Projective Planes

Abstract

We study the quotients for the diagonal action of ${SL}_3(\mathbb{C})$ on the $n$-fold product of ${\mathbb{P}}^2(\mathbb{C})$: we are interested in describing how the quotient changes when we vary the polarization (i.e. the choice of an ample linearized line bundle). We illustrate the different techniques for the construction of a quotient, in particular the numerical criterion for semi-stability and the “elementary transformations” which are resolutions of precisely described singularities (case $n=6$).