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enter image description here

I am reading 'An invitation to Quantum cohomology by J.Kock, I.vainsencher'.

I added a picture of the page on which I have question.

The projection is flat and therefore has positive relative (fiber) dimension.

How is this fact related that the direct image is zero?

Should I just consider intersection theory?

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    $\begingroup$ It would help if you include more context instead of embedding a scanned excerpt from a book. $\endgroup$
    – YCor
    Commented Aug 7, 2017 at 11:23
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    $\begingroup$ The proper pushforward map in homology, resp. Chow groups, is a graded homomorphism that preserves the homological degree, i.e., $H_q(\overline{M}_{0,n}\times \mathbb{P}^r)\to H_q(\mathbb{P}^r)$, resp. $\text{CH}_p(\overline{M}_{0,n}\times \mathbb{P}^r)\to \text{CH}_p(\mathbb{P}^r)$. The fundamental class of $\overline{M}_{0,n}\times \mathbb{P}^r$ is in degree $2(r+(n-3))$, resp. in $\text{CH}_{r+(n-3)}(\overline{M}_{0,n}\times \mathbb{P}^r)$. Since the homology groups, resp. Chow groups, of $\mathbb{P}^r$ vanish in degrees $>2r$, resp. $>r$, the pushforward is nonzero if and only if $n=3$. $\endgroup$
    – Jason Starr
    Commented Aug 7, 2017 at 11:42

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