3.1 Adjunction
Here we want to follow Hartshorne.
Suppose \(k\) is an algebraically closed field and \(Y\) is a codimension \(r\) nonsingular subvariety of a nonsingular variety \(X\) over \(k\) with normal bundle \(N\). Let \(\omega _Y\) and \(\omega _X\) denote the dualizing sheaves of \(X\) and \(y\) respectively. Then \(\omega _Y = \omega _X\otimes _{\mathcal{O}_X}\wedge ^r N\).
In the case \(r=1\), \(Y\) is a prime Cartier divisor on \(X\) with \(L=\mathcal{O}_X(Y)\) a line bundle and \(N=L_{|Y} = L\otimes _{\mathcal{O}_X}\mathcal{O}_Y\). So we get \(\omega _Y= \omega _X\otimes _{\mathcal{O}_X} L_{|Y}\).
This is [ 4 , Proposition 8.20 ] .
The main point is that, if \(I\) denotes the idea sheaf of \(Y\), then we get an exact sequence
with \(I/I^2 = N^*\), the conormal bundle on \(X\).
In the case \(r=1\), \(I=\mathcal{O}_X(-Y)=L^{-1}\).
If \(X\) is a smooth variety, \(\omega _X\) is a line bundle. So it corresponds to a Cartier divisor \(K_X\). Of course, \(K_X\) isn’t unique as an element of \(\operatorname{\mathrm{Div}}X\), but its linear equivalence class is unique. So we just pick a divisor \(K_X\) and, by abuse of notation call it the canonical divisor of \(X\).
Suppose \(X\) is a smooth surface and \(C\) is a smooth curve of genus \(g\) on \(X\). Then \(2g-2 = C\cdot (C+K)\) or, equivalently,
This is [ 4 , § V.1 ] .
We have \(2g-2 = \deg \omega _Y = \deg (\omega _X\otimes \mathcal{O}_X(C))_{|C} = (K+C)\cdot C\).