I've learnt that if $Y\subset X$ is a smooth subvariety of codimension 1 then the degree of canonical divisors of $Y$ and $X$ are related by

\begin{equation} \deg K_Y = \deg((K_X + Y)\cap Y). \end{equation}

However, I'm wondering if I can say more about this relationship. In particular, I have seen the following on Wikipedia (https://en.wikipedia.org/wiki/Adjunction_formula)

\begin{equation} K_Y = (K_X + Y)|_Y \end{equation}

which led me to believe that maybe,

\begin{equation} K_Y = (K_X + Y)\cap Y ? \end{equation}

Where $\cap$ is an intersection product and both sides are equal as divisor classes in $Div(Y)$.

We also know that if $i:Y\rightarrow X$ is an inclusion then $\mathcal{O}_Y(K_Y) = i^*(\mathcal{O}_X(K_X)\otimes \mathcal{O}_X(Y))$. I naively would think that by taking chern class on both sides, we have \begin{align} K_Y &= c_1(\mathcal{O}_Y(K_Y)) = c_1(i^*(\mathcal{O}_X(K_X)\otimes \mathcal{O}_X(Y)))\\ &= i^{-1}c_1(\mathcal{O}_X(K_X)\otimes \mathcal{O}_X(Y)) \\ &= i^{-1}(K_X + Y) \\ &= (K_X + Y)\cap Y \end{align}

which is probably wrong since I don't know much intersection theory. So could someone help me, please? Thank you.


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