First Chern class of line bundle corresponding to divisor

Question

If I know an effective divisor $D$, then there is a line bundle $L_D$ corresponding to this divisor. How can I compute the first Chern class of $L_D$? For example, on $\mathbb{C}\mathbb{P}^3$, $D=[F]$, where $F=z_0z_1^2+z_2z_3(z_2-z_3)$. I am a newcomer of algebraic geometry, so some reference or details would be welcomed.

Answer 1

Let's say $X$ is a smooth projective variety for simplicity.

The short, maybe unhelpful answer is this: the first Chern class of $L_D$ is exactly the element $[D]$ of the Picard group $\operatorname{Pic}(X)$ canonically defined by the divisor $D$.

To say something more informative, one has to say what one wants to compute $c_1(L_D)$ in terms of. In your example, $\operatorname{Pic}(\mathbf P^3)$ is isomorphic to $\mathbf Z$, with the isomorphism taking a divisor to its degree. (That isn't 100% obvious: there is a detailed proof in Shafarevich.) So the Picard group is generated by $[H]$, the class of a plane.

Your $D$ is a cubic (degree 3) hypersurface in $\mathbf P^3$, so its class in the Picard group is $[D]=3[H]$. (If you want to phrase this in terms of the line bundle, you can say $c_1(L_D)=3[H]$.)

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For a general $X$, the situation will be more complicated. The Picard group $\operatorname{Pic}(X)$ might not be generated by a single element, or it might even not be finitely generated. (Example: an elliptic curve.) One way to deal with this is to work in the Néron–Severi group $NS(X)$ instead. Roughly speaking, $NS(X)$ only remembers the numerical properties of a divisor class or line bundle — in other words, its intersection numbers with curves in $X$. The great advantage is that it is always finitely generated (the so-called Theorem of the Base.) Also there is a quotient map $\operatorname{Pic}(X) \rightarrow NS(X)$, so every divisor or line bundle has a well-defined class in $NS(X)$ too.

Now suppose you have fixed a basis of $NS(X)$. To calculate the class of $D$ or $L_D$ in $NS(X)$ in terms of that basis, one calculates the intersection numbers $D \cdot C$ for curves $C$ in $X$. That leads to a system of linear equations for the coefficients of $D$ in the chosen basis, and solving the system tells you the (numerical) class of $D$ in terms of the chosen basis.