# Fibers and Push-forward

Let $$f\colon X\longrightarrow Y$$ be a surjective morphism of projective varieties. Consider a non-singular irreducible projective curve $$T$$ and surjective morphism $$p_Y\colon Y\longrightarrow T$$. Then we get the following commutative diagram $$\require{AMScd}$$ $$\begin{CD} X_t @>f_t>> Y_t\\ @V V V @VV V\\ X @>f>> Y\\ @Vp_X V V @Vp_YV V\\ T @>>\mathsf{id}_T> T \end{CD}$$ so in particular an induced map between fibers $$f_t\colon X_t\longrightarrow Y_t$$. I would to check the following result:

Result: Let $$V\subseteq X$$ be an irreducible projective variety such that $$\dim V=\dim W$$, where $$W:=f(V)$$. Denote by $$[V_t]$$ the cycle associated to the projective variety $$p_X^{-1}(t)\cap V$$ and by $$f_\ast$$ the push-forward induced by $$f$$. Then $$(\#) \: \: \: \:(f_t)_\ast([V_t])=\big[f_\ast[V]\big]_t=\deg(f_{|V})[W_t].$$ This is a well know formula (eg. Prop. 10.1 of [Ful]), but in the reference I cited the Author proves it in a different way. Indeed, he consider intersection fibers instead of cycle associated to scheme theoretic fibers as I am doing.

My Attempt: We can assume that $$V$$ and $$Y$$ are affine and $$f$$ is dominant. Denote the coordinate rings of the fibers $$V_t,W_t$$ by $$A:=k[V_t]$$ and $$R=k[W_t]$$. Then we have $$(f_t)_\ast[V_t]=\sum_S\ell(A_{\mathfrak q_S})\deg({f_t}_{|S})f_t(S),$$ where $$S$$ runs over the irreducible components of $$V_t$$ such that $$\dim f(S)=dim W-1$$. On the other hand, we have $$\deg(f_{|V})\cdot[W_t]=[k(V):k(W)]\cdot\sum_Z\ell(R_{\mathfrak p_Z})Z,$$ where $$Z$$ runs over the irreducible components of $$W_t$$. Now fix an irreducible component $$Z$$ (corresponding to the minimal prime $$\mathfrak p\subset R$$) and consider the irreducible components $$S_1,\ldots,S_n$$ of $$V_t$$ (corresponding to the minimal primes $$\mathfrak q_1,\ldots \mathfrak q_n\subset A$$) whose image via $$f_t$$ is just $$Z$$. Proving $$(\#)$$ is then equivalent to provving the following formula $$[k(V):k(W)]\cdot\ell(R_\mathfrak p)=\sum_{i=1}^n[k(\mathfrak q_i):k(\mathfrak p)]\cdot\ell(A_{\mathfrak q_i}).$$

A similar aproach can be found here. More precisely, this lemma should be the key-point of the proof.

Add: The proof of this result would be important because one could check that the push-forward preserves algebraic equivalence of cycles by considering the following commutative diagram:

$$\begin{CD} \lbrace t\rbrace\times X @>f>> \lbrace t \rbrace\times Y\\ @V V V @VV V\\ T\times X @>\mathsf{id}\times f>> T\times Y\\ @Vp_X V V @Vp_YV V\\ T @>>\mathsf{id}_T> T \end{CD}$$

Thanks in advance for any help.

[Ful] Fulton, W. (1988) Intersection Theory. New York: Springer-Verlag.

## 1 Answer

Let me answer to my own question. At least, if someone will struggle with my same proplem in the future he will can find a solution.

The formula I was tempting to prove is actually true, with some modification. More precisely, in the Handbook of K-Theory (ed. Grayson and Friedlander), Thomas Geisser, in his paper Motivic Cohomology, K-Theory and Topological Cyclic Homology proves the following result. By mean of standard geometric arguments, He deduces that it is enough to check the claim in the affine case and consider $$V$$ in place of $$X$$ and $$f(V)$$ in place of $$Y$$. Thus, the proposition is proved by showing the formula 