Observability in invariant theory II: Divisors and rational invariants

Renner, Lex

doi:10.1090/proc/12804

Observability in invariant theory II: Divisors and rational invariants
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Proc. Amer. Math. Soc. 143 (2015), 4113-4121 Request permission

Abstract:

Let $G\times X\to X$ be an action of the connected algebraic group $G$ on the irreducible, affine variety $X$. We discuss the relationship between $[k[X]^G]$ and $k(X)^G$, where $[k[X]^G]$ denotes the quotient field of $k[X]^G$. We are particularly interested in the following three questions. (1) When is the inclusion $[k[X]^G]\subseteq k(X)^G$ a finite extension of fields? (2) What is the role of $G$-invariant divisors? (3) What is the exact characterization of “observable in codimension one”?

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