The Stacks project

Lemma 43.17.1. Let $X$ be a nonsingular variety. Let $a, b \in \mathbf{P}^1$ be distinct closed points. Let $k \geq 0$.

  1. If $W \subset X \times \mathbf{P}^1$ is a closed subvariety of dimension $k + 1$ which intersects $X \times a$ properly, then

    1. $[W_ a]_ k = W \cdot X \times a$ as cycles on $X \times \mathbf{P}^1$, and

    2. $[W_ a]_ k = \text{pr}_{X, *}(W \cdot X \times a)$ as cycles on $X$.

  2. Let $\alpha $ be a $(k + 1)$-cycle on $X \times \mathbf{P}^1$ which intersects $X \times a$ and $X \times b$ properly. Then $pr_{X,*}( \alpha \cdot X \times a - \alpha \cdot X \times b)$ is rationally equivalent to zero.

  3. Conversely, any $k$-cycle which is rationally equivalent to $0$ is of this form.

Proof. First we observe that $X \times a$ is an effective Cartier divisor in $X \times \mathbf{P}^1$ and that $W_ a$ is the scheme theoretic intersection of $W$ with $X \times a$. Hence the equality in (1)(a) is immediate from the definitions and the calculation of intersection multiplicity in case of a Cartier divisor given in Lemma 43.16.4. Part (1)(b) holds because $W_ a \to X \times \mathbf{P}^1 \to X$ maps isomorphically onto its image which is how we viewed $W_ a$ as a closed subscheme of $X$ in Section 43.8. Parts (2) and (3) are formal consequences of part (1) and the definitions. $\square$

Comments (0)

There are also:

  • 2 comment(s) on Section 43.17: Intersection product using Tor formula

Post a comment

Your email address will not be published. Required fields are marked.

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 0B07. Beware of the difference between the letter 'O' and the digit '0'.