Return to Answer

replace bullet by list for easier referencing

Source Link

For $(1) \iff (3)$, we have the following chain of identities:

$\operatorname{trdeg} \operatorname{Frac} \left( \bigoplus_{m ≥ 0} H^0(X, mL) \right)$ is equal to

$\max \operatorname{trdeg}(F)$ where $F$ is a finitely generated subfield of $ \operatorname{Frac} \left( \bigoplus_{m ≥ 0} H^0(X, mL) \right)$, which is equal to

$\max \operatorname{trdeg}( \operatorname{Frac}(R))$ where $R$ is a finitely generated subring of $\bigoplus_{m ≥ 0} H^0(X, mL) $ (take $R$ generated by the numerators and denominators of $F$ ), which is equal to

$\max \operatorname{trdeg}( \operatorname{Frac}(R))$, where $R$ is the subring generated by $\bigoplus_{m \in S} H^0(X, mL) $ for $S$ a finite set of natural numbers (take the supports of the generators), which is equal to

$\max \operatorname{trdeg}( \operatorname{Frac}(R))$, where $R$ is the subring generated by $ H^0(X, nL) $ (take the least common multiple of $S$, and note the old generators are finite over this ring), which is equal to

the max of the dimension of the spectrum of $R$, where $R$ is the subring generated by $ H^0(X, nL) $, which is equal to

the max of the dimension of the affine cone on the closure of the image of the rational map $φ_{|nL|} : X \dashrightarrow \mathbb{P}(H^0(X, n L)^*)$ (since the spectrum is equal to that cone)

the max of the dimension of the image of the rational map $φ_{|nL|} : X \dashrightarrow \mathbb{P}(H^0(X, n L)^*)$, plus 1 (since the affine cone has dimension one higher).

$\operatorname{trdeg} \operatorname{Frac} \left( \bigoplus_{m ≥ 0} H^0(X, mL) \right)$ is equal to

$\max \operatorname{trdeg}(F)$ where $F$ is a finitely generated subfield of $ \operatorname{Frac} \left( \bigoplus_{m ≥ 0} H^0(X, mL) \right)$, which is equal to

$\max \operatorname{trdeg}( \operatorname{Frac}(R))$ where $R$ is a finitely generated subring of $\bigoplus_{m ≥ 0} H^0(X, mL) $ (take $R$ generated by the numerators and denominators of $F$ ), which is equal to

$\max \operatorname{trdeg}( \operatorname{Frac}(R))$, where $R$ is the subring generated by $\bigoplus_{m \in S} H^0(X, mL) $ for $S$ a finite set of natural numbers (take the supports of the generators), which is equal to

$\max \operatorname{trdeg}( \operatorname{Frac}(R))$, where $R$ is the subring generated by $ H^0(X, nL) $ (take the least common multiple of $S$, and note the old generators are finite over this ring), which is equal to

the max of the dimension of the spectrum of $R$, where $R$ is the subring generated by $ H^0(X, nL) $, which is equal to

the max of the dimension of the affine cone on the closure of the image of the rational map $φ_{|nL|} : X \dashrightarrow \mathbb{P}(H^0(X, n L)^*)$ (since the spectrum is equal to that cone)

the max of the dimension of the image of the rational map $φ_{|nL|} : X \dashrightarrow \mathbb{P}(H^0(X, n L)^*)$, plus 1 (since the affine cone has dimension one higher).

For $(1) \iff (3)$, we have the following chain of identities:

$\operatorname{trdeg} \operatorname{Frac} \left( \bigoplus_{m ≥ 0} H^0(X, mL) \right)$ is equal to

$\max \operatorname{trdeg}(F)$ where $F$ is a finitely generated subfield of $ \operatorname{Frac} \left( \bigoplus_{m ≥ 0} H^0(X, mL) \right)$, which is equal to

$\max \operatorname{trdeg}( \operatorname{Frac}(R))$ where $R$ is a finitely generated subring of $\bigoplus_{m ≥ 0} H^0(X, mL) $ (take $R$ generated by the numerators and denominators of $F$ ), which is equal to

$\max \operatorname{trdeg}( \operatorname{Frac}(R))$, where $R$ is the subring generated by $\bigoplus_{m \in S} H^0(X, mL) $ for $S$ a finite set of natural numbers (take the supports of the generators), which is equal to

$\max \operatorname{trdeg}( \operatorname{Frac}(R))$, where $R$ is the subring generated by $ H^0(X, nL) $ (take the least common multiple of $S$, and note the old generators are finite over this ring), which is equal to

the max of the dimension of the spectrum of $R$, where $R$ is the subring generated by $ H^0(X, nL) $, which is equal to

the max of the dimension of the affine cone on the closure of the image of the rational map $φ_{|nL|} : X \dashrightarrow \mathbb{P}(H^0(X, n L)^*)$ (since the spectrum is equal to that cone)

the max of the dimension of the image of the rational map $φ_{|nL|} : X \dashrightarrow \mathbb{P}(H^0(X, n L)^*)$, plus 1 (since the affine cone has dimension one higher).

For $(1) \iff (3)$, we have the following chain of identities:

$\operatorname{trdeg} \operatorname{Frac} \left( \bigoplus_{m ≥ 0} H^0(X, mL) \right)$ is equal to

$\max \operatorname{trdeg}(F)$ where $F$ is a finitely generated subfield of $ \operatorname{Frac} \left( \bigoplus_{m ≥ 0} H^0(X, mL) \right)$, which is equal to

$\max \operatorname{trdeg}( \operatorname{Frac}(R))$ where $R$ is a finitely generated subring of $\bigoplus_{m ≥ 0} H^0(X, mL) $ (take $R$ generated by the numerators and denominators of $F$ ), which is equal to

$\max \operatorname{trdeg}( \operatorname{Frac}(R))$, where $R$ is the subring generated by $\bigoplus_{m \in S} H^0(X, mL) $ for $S$ a finite set of natural numbers (take the supports of the generators), which is equal to

$\max \operatorname{trdeg}( \operatorname{Frac}(R))$, where $R$ is the subring generated by $ H^0(X, nL) $ (take the least common multiple of $S$, and note the old generators are finite over this ring), which is equal to

the max of the dimension of the spectrum of $R$, where $R$ is the subring generated by $ H^0(X, nL) $, which is equal to

the max of the dimension of the affine cone on the closure of the image of the rational map $φ_{|nL|} : X \dashrightarrow \mathbb{P}(H^0(X, n L)^*)$ (since the spectrum is equal to that cone)

the max of the dimension of the image of the rational map $φ_{|nL|} : X \dashrightarrow \mathbb{P}(H^0(X, n L)^*)$, plus 1 (since the affine cone has dimension one higher).

Source Link

created Feb 24, 2022 at 0:26

For $(1) \iff (3)$, we have the following chain of identities:

$\operatorname{trdeg} \operatorname{Frac} \left( \bigoplus_{m ≥ 0} H^0(X, mL) \right)$ is equal to
$\max \operatorname{trdeg}(F)$ where $F$ is a finitely generated subfield of $ \operatorname{Frac} \left( \bigoplus_{m ≥ 0} H^0(X, mL) \right)$, which is equal to
$\max \operatorname{trdeg}( \operatorname{Frac}(R))$ where $R$ is a finitely generated subring of $\bigoplus_{m ≥ 0} H^0(X, mL) $ (take $R$ generated by the numerators and denominators of $F$ ), which is equal to
$\max \operatorname{trdeg}( \operatorname{Frac}(R))$, where $R$ is the subring generated by $\bigoplus_{m \in S} H^0(X, mL) $ for $S$ a finite set of natural numbers (take the supports of the generators), which is equal to
$\max \operatorname{trdeg}( \operatorname{Frac}(R))$, where $R$ is the subring generated by $ H^0(X, nL) $ (take the least common multiple of $S$, and note the old generators are finite over this ring), which is equal to
the max of the dimension of the spectrum of $R$, where $R$ is the subring generated by $ H^0(X, nL) $, which is equal to
the max of the dimension of the affine cone on the closure of the image of the rational map $φ_{|nL|} : X \dashrightarrow \mathbb{P}(H^0(X, n L)^*)$ (since the spectrum is equal to that cone)
the max of the dimension of the image of the rational map $φ_{|nL|} : X \dashrightarrow \mathbb{P}(H^0(X, n L)^*)$, plus 1 (since the affine cone has dimension one higher).