# The transcendence degree of the algebras of invariants

Let $V_n,V_m$ be the vector $\mathbb{C}$-spaces of the binary forms of degrees $n,m$ considered as usual $SL_2$-modules. Let $I_{n,m}=\mathbb{C}[V_n \oplus V_m]^{SL_2}$ and $C_{n,m}=\mathbb{C}[V_n \oplus V_m \oplus \mathbb{C}^2]^{SL_2}$ be the corresponding algebras of joint invariants and covariants. The transcendence degree of $I_{n,m}$ over $\mathbb{C}$ is $n+m-1.$

Questions:

1. What is the transcendence degree of $C_{n,m}$ over the subalgebra $I_{n,m}$?

2. What is the transcendence degree of $C_{n,m}$ over the quotient field of the subalgebra $I_{n,m}$?

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I think you meant to write that the transcendence degree of $I_{n,m}$ (not $C_{n,m}$) over $\mathbb{C}$ is $m{+}n{-}1$. Isn't it true that the transcendence degree of $C_{n,m}$ over $I_{n,m}$ is $2$? –  Robert Bryant Dec 1 '12 at 13:35
@Robert Thanks, corrected. Please, any reference to the result that $trdegC_{n,m}$ over $I_{n,m}$ is 2. What about the second question? –  Melania Dec 3 '12 at 6:38