Let $ f: \mathbb{P}^n \dashrightarrow (\mathbb{P}^1)^n , [x\_0:\dots :x\_n] \mapsto ([x\_0,x\_1],[x\_0,x\_2], \dots ,[x\_0,x\_n])$ a birational map.
Is $f$ a Crepant birational map?



Crepant Birational Map
Kontsevich proved in '95 that the hodge polynomial is independent of crepant resolution over a field of characteristic zero. He created the field of motivic integration to prove this fact. Here are the respective hodge polynomials of the two spaces in question where I repress the fact that we are working over the field $k$ just for notational simplicity:


$$H([\mathbb{P}^n], u,v) = \sum\_{i=0}^{n} H([\mathbb{A}^i], u,v) =\sum\_{i=0}^{n} (uv)^i $$


and


$$H([\prod\_{i=1}^{n}\mathbb{P}^{1}], u,v) = \prod\_{i=1}^{n} (H([\mathbb{A}^1], u,v)+1) = \prod\_{i=1}^{n} (uv+1) = (uv+1)^n = \sum\_{i=0}^{n} {n \choose i} (uv)^i $$


where we denote by $[X]$ the equivalence class of the variety $X$ in the grothendieck ring of varieties. Therefore, there are no crepant birational maps between $\mathbb{P}^n$ and $\prod\_{i=1}^{n} \mathbb{P}^1$. 


Note, actually, here $[X]$ lives in the image of the Grothendieck ring of varieties localized at $[\mathbb{A}^1]$ under the completion homomorphism given by the dimension filtration. But, the hodge polynomial ring homomorphism
`'$$H(X,u,v) = \sum\_{i,j} (-1)^{i+j} dim(H^i(X, \Omega\_{X}^{j}))u^iv^j $$'
factors through the ring homomorphism $X \mapsto [X]$. 


I am only sure that this argument works over a field of characteristic zero for two reasons: 
1) there is not currently a resolution of singularities over fields of positive characteristic, and 
2) I don't know how to define the hodge polynomial in positive characteristic. 


Note that this result is sometimes known as Batyrev's Theorem because he originally proved a similar result in '95 involving the beta numbers via p-adic integration and the weil conjectures. 


