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As you note $M$ is locally a complete intersection. Take a point and the corresponding local equations near that point, extend those equations to $\mathbb P^m$ so you have a complete intersection subvariety $X\subset \mathbb P^m$ of codimension $r$ such that $X=M\cup N$ for some other subvariety $N\subset \mathbb P^m$ of codimension $r$.

Generally both $X$ and $N$ are singular, but since $X$ is a complete intersection it is Gorenstein and its canonical bundle is computed the same way as in the smooth case, so $$ \omega_X\simeq \mathscr O_{\mathbb P^m}(d-m-1)|_X\tag{$\star$} $$ where $d$ is the sum of the degree of equations defining $X$.

Claim $\quad$ $\omega_M\subseteq \mathscr O_{\mathbb P^m}(d-m-1)$ where $d$ is the minimum of the sum of degrees of local defining equations for $M$.

Notation: For a subvariety $Z\subseteq \mathbb P^m$, denote the ideal sheaf of $Z$ by $\mathscr I_Z$.

Remark: Note that this claim does not require that $\omega_M$ is a line bundle restricted from $\mathbb P^m$. See also the corollary.

Proof: Since $M\subseteq X$, we have $\mathscr I_X\subseteq \mathscr I_M$ and hence we get a natural morphism $$ \iota: (\mathscr I_X/\mathscr I_X^2)|_M \to \mathscr I_M/\mathscr I_M^2. $$

As $X$ is a complete intersection, $\mathscr I_X/\mathscr I_X^2$ is locally free, and hence in particular it is torsion-free. Furthermore, $\iota$ is an isomorphism on $M\setminus N$ which is an open dense subset of $M$. It follows that $\ker\iota$ is a torsion subsheaf of $\mathscr I_X/\mathscr I_X^2$, so $\ker\iota=0$ and thus $\iota$ is an injection on all of $M$.

As $M$ is smooth,
$$ \det\mathscr N_{M/\mathbb P^m}=(\det \mathscr I_M/\mathscr I_M^2)^* $$ is a line bundle, so taking duals and determinants we get that $$ \det\mathscr N_{M/\mathbb P^m}\subseteq \det\mathscr N_{X/\mathbb P^m}|_M. $$

By applying the adjunction formula for both $X\subset\mathbb P^m$ and $M\subset\mathbb P^m$, we get hat $$\omega_M\subseteq \omega_X|_M.$$ The Claim follows by $(\star)$. $\quad\square$

Corollary $\quad$ If $\omega_M$ is a line bundle restricted from $\mathbb P^m$, then $\omega_M\simeq \mathscr O_{\mathbb P^m}(q)$ for some $q\leq d-m-1$ where $d$ is as above.

Finally, a note on the notion of "subcanonical". I think by that some people mean a singularity that formally satisfies the definition of a canonical singularity, but with a not necessarily effective divisor as boundary.

As you note $M$ is locally a complete intersection. Take a point and the corresponding local equations near that point, extend those equations to $\mathbb P^m$ so you have a complete intersection subvariety $X\subset \mathbb P^m$ of codimension $r$ such that $X=M\cup N$ for some other subvariety $N\subset \mathbb P^m$ of codimension $r$.

Generally both $X$ and $N$ are singular, but since $X$ is a complete intersection it is Gorenstein and its canonical bundle is computed the same way as in the smooth case, so $$ \omega_X\simeq \mathscr O_{\mathbb P^m}(d-m-1)|_X\tag{$\star$} $$ where $d$ is the sum of the degree of equations defining $X$.

Claim $\quad$ $\omega_M\subseteq \mathscr O_{\mathbb P^m}(d-m-1)|_M$ where $d$ is the minimum of the sum of degrees of local defining equations for $M$.

Notation: For a subvariety $Z\subseteq \mathbb P^m$, denote the ideal sheaf of $Z$ by $\mathscr I_Z$.

Remark: Note that this claim does not require that $\omega_M$ is a line bundle restricted from $\mathbb P^m$. See also the corollary.

Proof: Since $M\subseteq X$, we have $\mathscr I_X\subseteq \mathscr I_M$ and hence we get a natural morphism $$ \iota: (\mathscr I_X/\mathscr I_X^2)|_M \to \mathscr I_M/\mathscr I_M^2. $$

As $X$ is a complete intersection, $\mathscr I_X/\mathscr I_X^2$ is locally free, and hence in particular it is torsion-free. Furthermore, $\iota$ is an isomorphism on $M\setminus N$ which is an open dense subset of $M$. It follows that $\ker\iota$ is a torsion subsheaf of $\mathscr I_X/\mathscr I_X^2$, so $\ker\iota=0$ and thus $\iota$ is an injection on all of $M$.

As $M$ is smooth,
$$ \det\mathscr N_{M/\mathbb P^m}=(\det \mathscr I_M/\mathscr I_M^2)^* $$ is a line bundle, so taking duals and determinants we get that $$ \det\mathscr N_{M/\mathbb P^m}\subseteq \det\mathscr N_{X/\mathbb P^m}|_M. $$

By applying the adjunction formula for both $X\subset\mathbb P^m$ and $M\subset\mathbb P^m$, we get hat $$\omega_M\subseteq \omega_X|_M.$$ The Claim follows by $(\star)$. $\quad\square$

Corollary $\quad$ If $\omega_M$ is a line bundle restricted from $\mathbb P^m$, then $\omega_M\simeq \mathscr O_{\mathbb P^m}(q)$ for some $q\leq d-m-1$ where $d$ is as above.

Finally, a note on the notion of "subcanonical". I think by that some people mean a singularity that formally satisfies the definition of a canonical singularity, but with a not necessarily effective divisor as boundary.

As you note $M$ is locally a complete intersection. Take a point and the corresponding local equations near that point, extend those equations to $\mathbb P^m$ so you have a complete intersection subvariety $X\subset \mathbb P^m$ of codimension $r$ such that $X=M\cup N$ for some other subvariety $N\subset \mathbb P^m$ of codimension $r$.

Generally both $X$ and $N$ are singular, but since $X$ is a complete intersection it is Gorenstein and its canonical bundle is computed the same way as in the smooth case.

If $N$ is a Cartier divisor in $X$, then you have the following relationship: $$ \omega_M\simeq \omega_X(-N)|_M. $$ This can be proven by applying the adjunction formula for both $X\subset\mathbb P^m$ and $M\subset\mathbb P^m$. Since $M$ and $N$ have no common components, we know that $N|_M$ is effective, so this implies that $$\omega_M\subseteq \omega_X|_M.$$ and hence $$\omega_M=\mathscr O(d-m-1)\tag{$\star$}$$ where $d$ is at most the minimum value for the sum of degrees of equations defining $M$ as a local complete intersection.

I believe that even if $N$ is not Cartier in $X$ one can say something similar, but it is more complicated. Adding your assumption that $\omega_M$ is a restriction of a line bundle from $\mathbb P^m$ and hence from $X$ should imply that actually $N$ is Cartier in a neighbourhood of $M\cap N$ in $X$, so very likely you have $(\star)$ in that case as well.

Finally a note on the notion of "subcanonical". I think by that some people mean a singularity that formally satisfies the definition of a canonical singularity, but with a not necessarily effective divisor as boundary.

As you note $M$ is locally a complete intersection. Take a point and the corresponding local equations near that point, extend those equations to $\mathbb P^m$ so you have a complete intersection subvariety $X\subset \mathbb P^m$ of codimension $r$ such that $X=M\cup N$ for some other subvariety $N\subset \mathbb P^m$ of codimension $r$.

Generally both $X$ and $N$ are singular, but since $X$ is a complete intersection it is Gorenstein and its canonical bundle is computed the same way as in the smooth case, so $$ \omega_X\simeq \mathscr O_{\mathbb P^m}(d-m-1)|_X\tag{$\star$} $$ where $d$ is the sum of the degree of equations defining $X$.

Claim $\quad$ $\omega_M\subseteq \mathscr O_{\mathbb P^m}(d-m-1)$ where $d$ is the minimum of the sum of degrees of local defining equations for $M$.

Notation: For a subvariety $Z\subseteq \mathbb P^m$, denote the ideal sheaf of $Z$ by $\mathscr I_Z$.

Remark: Note that this claim does not require that $\omega_M$ is a line bundle restricted from $\mathbb P^m$. See also the corollary.

Proof: Since $M\subseteq X$, we have $\mathscr I_X\subseteq \mathscr I_M$ and hence we get a natural morphism $$ \iota: (\mathscr I_X/\mathscr I_X^2)|_M \to \mathscr I_M/\mathscr I_M^2. $$

As $X$ is a complete intersection, $\mathscr I_X/\mathscr I_X^2$ is locally free, and hence in particular it is torsion-free. Furthermore, $\iota$ is an isomorphism on $M\setminus N$ which is an open dense subset of $M$. It follows that $\ker\iota$ is a torsion subsheaf of $\mathscr I_X/\mathscr I_X^2$, so $\ker\iota=0$ and thus $\iota$ is an injection on all of $M$.

As $M$ is smooth,
$$ \det\mathscr N_{M/\mathbb P^m}=(\det \mathscr I_M/\mathscr I_M^2)^* $$ is a line bundle, so taking duals and determinants we get that $$ \det\mathscr N_{M/\mathbb P^m}\subseteq \det\mathscr N_{X/\mathbb P^m}|_M. $$

By applying the adjunction formula for both $X\subset\mathbb P^m$ and $M\subset\mathbb P^m$, we get hat $$\omega_M\subseteq \omega_X|_M.$$ The Claim follows by $(\star)$. $\quad\square$

Corollary $\quad$ If $\omega_M$ is a line bundle restricted from $\mathbb P^m$, then $\omega_M\simeq \mathscr O_{\mathbb P^m}(q)$ for some $q\leq d-m-1$ where $d$ is as above.

Finally, a note on the notion of "subcanonical". I think by that some people mean a singularity that formally satisfies the definition of a canonical singularity, but with a not necessarily effective divisor as boundary.

As you note $M$ is locally a complete intersection. Take a point and the corresponding local equations near that point, extend those equations to $\mathbb P^m$ so you have a complete intersection subvariety $X\subset \mathbb P^m$ of codimension $r$ such that $X=M\cup N$ for some other subvariety $N\subset \mathbb P^m$ of codimension $r$.

Generally both $X$ and $N$ are singular, but since $X$ is a complete intersection it is Gorenstein and its canonical bundle is computed the same way as in the smooth case.

If $N$ is a Cartier divisor in $X$, then you have the following relationship: $$ \omega_M\simeq \omega_X(-N)|_M. $$ This can be proven by applying the adjunction formula for both $X\subset\mathbb P^m$ and $M\subset\mathbb P^m$. Since $M$ and $N$ have no common components, we know that $N|_M$ is effective, so this implies that $$\omega_M\subseteq \omega_X|_M.\tag{$\star$}$$

I believe that even if $N$ is not Cartier in $X$ one can say something similar, but it is more complicated. Adding your assumption that $\omega_M$ is a restriction of a line bundle from $\mathbb P^m$ and hence from $X$ should imply that actually $N$ is Cartier in a neighbourhood of $M\cap N$ in $X$, so very likely you have $(\star)$ in that case as well.

Finally a note on the notion of "subcanonical". I think by that some people mean a singularity that formally satisfies the definition of a canonical singularity, but with a not necessarily effective divisor as boundary.

As you note $M$ is locally a complete intersection. Take a point and the corresponding local equations near that point, extend those equations to $\mathbb P^m$ so you have a complete intersection subvariety $X\subset \mathbb P^m$ of codimension $r$ such that $X=M\cup N$ for some other subvariety $N\subset \mathbb P^m$ of codimension $r$.

Generally both $X$ and $N$ are singular, but since $X$ is a complete intersection it is Gorenstein and its canonical bundle is computed the same way as in the smooth case.

If $N$ is a Cartier divisor in $X$, then you have the following relationship: $$ \omega_M\simeq \omega_X(-N)|_M. $$ This can be proven by applying the adjunction formula for both $X\subset\mathbb P^m$ and $M\subset\mathbb P^m$. Since $M$ and $N$ have no common components, we know that $N|_M$ is effective, so this implies that $$\omega_M\subseteq \omega_X|_M.$$ and hence $$\omega_M=\mathscr O(d-m-1)\tag{$\star$}$$ where $d$ is at most the minimum value for the sum of degrees of equations defining $M$ as a local complete intersection.

I believe that even if $N$ is not Cartier in $X$ one can say something similar, but it is more complicated. Adding your assumption that $\omega_M$ is a restriction of a line bundle from $\mathbb P^m$ and hence from $X$ should imply that actually $N$ is Cartier in a neighbourhood of $M\cap N$ in $X$, so very likely you have $(\star)$ in that case as well.

Finally a note on the notion of "subcanonical". I think by that some people mean a singularity that formally satisfies the definition of a canonical singularity, but with a not necessarily effective divisor as boundary.