A correspondence of good G-sets under partial geometric quotients

2016/11/09 by Johannes Schmitt, Schmitt, Johannes
Mathematics · #14L24 #14L30 #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG #msc:14L24 #msc:14L30

paper · pdf · doi:10.48550/arxiv.1611.02958

18 pages, comments welcome

arxiv created 2016/11/09 · arxiv updated 2016/11/10

Abstract

For a complex variety X with an action of a reductive group G and a geometric quotient π: X → X by a closed normal subgroup H ⊂ G, we show that open sets of X admitting good quotients by G= G / H correspond bijectively to open sets in X with good G-quotients. We use this to compute GIT-chambers and their associated quotients for the diagonal action of PGL₂ on (ℙ¹)ⁿ in certain subcones of the PGL₂-effective cone via a torus action on affine space. This allows us to represent these quotients as toric varieties with fans determined by convex geometry.

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