UMD 808J: Algebraic Surfaces

3.4.1 Proper push-forward

Suppose \(f:X\to Y\) is a proper morphism, and \(V\) is a \(k\)-dimensional closed subvariety of \(X\). Then the image \(W = f(V)\) is a closed subvariety of \(Y\) of dimension, and we get a field extension \(K(V)/K(W)\). Write \([V]\in Z_k X\) for the class of \(V\), and set

\begin{equation} f_*[V] := \begin{cases} [K(V):K(W)] [W], & \text{if } [K(V):K(W)] {\lt} \infty ;\\ 0, & \text{otherwise.} \end{cases} \end{equation}
3.17

Then, if \(\alpha = \sum _V n_V [V]\) is a cycle in \(Z_k X\), set \(f_*\alpha = \sum _V n_V f_*[V]\). In other words, extend \(f_*\) to a homomorphism \(f_*:Z_k X\to Z_k Y\) by linearity.

Proposition 3.20

Suppose \(f:X\to Y\) is a proper morphism. Then \(f_*(\operatorname{\mathrm{Rat}}_k X) \subseteq \operatorname{\mathrm{Rat}}_k Y\).

Proof ▼

This is not hard, but, for reasons of time, I think we should skip it. See  [ 3 , Proposition 1.4 ]

Definition 3.21

Suppose \(f:X\to Y\) is a proper morphism of separated schemes of finite type over a field \(k\), then the obvious homomorphism \(f_*: A_k X\to A_k Y\) coming from Proposition 3.20 is the proper pushforward on Chow groups.