Consider a nonsingular projective variety $X$ over an algebraically closed field $k$ and let $Y \subseteq X$ be a nonsingular closed subvariety. Let $\mathcal I \subseteq \mathcal O_X$ be the ideal sheaf of $Y$ and let $\mathcal L$ be a very ample invertible sheaf on $X$. If it makes the setup any easier we can assume that $X$ is projectively normal under the associated embedding to some projective space. Set $$ R(\mathcal L) := \bigoplus_{n \geq 0} \Gamma( X, \mathcal L^n ), $$ the section ring associated to $\mathcal L$. Then for each $m \geq 0$ we have the ideal $$ R_m(\mathcal L) := \bigoplus_{n \geq 0} \Gamma( X, \mathcal I^m \otimes_{\mathcal O_X} \mathcal L ) $$ of the ring $R(\mathcal L)$. These ideals define a decreasing multiplicative filtration on $R(\mathcal L)$ so we can consider the associated graded ring $gr R(\mathcal L)$.

Here is my question: Under what conditions can we conclude that this graded ring is Noetherian? Is this always true? If so, when can we conclude that this ring is generated in degree 1 over $k$? Unfortunately I do not have a deep knowledge of commutative algebra so I have gotten stuck on this point. If it makes the answer easier we can even assume that $Y$ is a point.

Consider a nonsingular projective variety $X$ over an algebraically closed field $k$ and let $Y \subseteq X$ be a nonsingular closed subvariety. Let $\mathcal I \subseteq \mathcal O_X$ be the ideal sheaf of $Y$ and let $\mathcal L$ be a very ample invertible sheaf on $X$. If it makes the setup any easier we can assume that $X$ is projectively normal under the associated embedding to some projective space. Set $$ R(\mathcal L) := \bigoplus_{n \geq 0} \Gamma( X, \mathcal L^n ), $$ the section ring associated to $\mathcal L$. Then for each $m \geq 0$ we have the homogeneous ideal $$ R_m(\mathcal L) := \bigoplus_{n \geq 0} \Gamma( X, \mathcal I^m \otimes_{\mathcal O_X} \mathcal L^n ) $$ of the ring $R(\mathcal L)$. These ideals define a decreasing multiplicative filtration on $R(\mathcal L)$ so we can consider the associated (bi-)graded ring $gr R(\mathcal L)$.

Here is my question: Under what conditions can we conclude that this graded ring is Noetherian? Is this always true? Unfortunately I do not have a deep knowledge of commutative algebra so I have gotten stuck on this point. If it makes the answer easier we can even assume that $Y$ is a point.