Definition 42.24.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. We define, for every integer $k$, an operation

$c_1(\mathcal{L}) \cap - : Z_{k + 1}(X) \to \mathop{\mathrm{CH}}\nolimits _ k(X)$

called intersection with the first Chern class of $\mathcal{L}$.

1. Given an integral closed subscheme $i : W \to X$ with $\dim _\delta (W) = k + 1$ we define

$c_1(\mathcal{L}) \cap [W] = i_*(c_1({i^*\mathcal{L}}) \cap [W])$

where the right hand side is defined in Definition 42.23.1.

2. For a general $(k + 1)$-cycle $\alpha = \sum n_ i [W_ i]$ we set

$c_1(\mathcal{L}) \cap \alpha = \sum n_ i c_1(\mathcal{L}) \cap [W_ i]$

There are also:

• 2 comment(s) on Section 42.24: Intersecting with an invertible sheaf

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 02SO. Beware of the difference between the letter 'O' and the digit '0'.