Timeline for Is there any relation between the pushforward of a divisor and the pushforward of its line bundle?

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Mar 28, 2022 at 8:06	comment	added	Francesco Polizzi		If $D$ is not effective, it seems to me that the result does not hold in general for birational contractions, since $f_\mathcal{L}(D)$ is not necessarily a line bundle. For instance, if $f \colon Y \to X$ is the blow-up of a surface $X$ at a smooth point $p$, with exceptional divisor $E$, then $$f_ \mathcal{L}(-nE) = \mathfrak{m}^n_p$$ for all $n \geq 1$.
Mar 27, 2022 at 18:08	comment	added	Will Sawin		(2) For a birational contraction, I think this is true whenever $f_* \mathcal L(D)$ is a line bundle. Take a (meromorphic) global section of $\mathcal L(D)$ with divisor $D$, which gives a (meromorphic) global section of the puhsforward, and note that its order of vanishing / pole at each codimension one point of $Y$ is the same as over the inverse image in $X$, since birational contractions are an isomorphism over codimension one points of the base.
Mar 27, 2022 at 18:07	comment	added	Will Sawin		(1) The Grothendieck-Riemann-Roch theorem gives an expression for the Chern classes of $f_* \mathcal L(D)$ in terms of $f_* D^n$ for various natural numbers $n$, in the case where the higher cohomology groups $R^i f_* \mathcal L(D)$ vanishes.
Mar 27, 2022 at 17:34	history	asked	Kim	CC BY-SA 4.0