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We survey the role played by GIT in the study of moduli spaces, with an emphasis on the birational geometry of GIT quotients.

We investigate the relationship between Geometric Invariant Theory (GIT) heights and weighted heights, with a focus on their interaction in weighted projective spaces and their application to binary forms. Building on the weighted height framework developed in previous papers, we relate it to Zhang's GIT height via the Veronese map. For a semistable cycle, we show that the GIT height decomposes into the logarithmic weighted height plus an Archimedean correction from the Chow metric.

Brown constructed a series of threefold flips given by the GIT quotient of a hypersurface in $\mathbb{C}^5$. In this article, we classify threefold flips and flops which are the GIT quotients of complete intersections in $\mathbb{C}^6$. We also show that there are no more new examples as GIT quotients of complete intersections in $\mathbb{C}^n$ with $n\geq7$.