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Let X be a Gorenstein variety. And $f:Y\longrightarrow X$ is a resolution, Y is normal. And we have $K_Y=f^*K_X+\sum_{i}a_iE_i$, $E_i$ are exceptional divisors. Then consider the push-forward of the line bundle $K_Y$ (since Y is Gorenstein, K_Y is locally free). $f_*K_Y=f_*f^*K_X+\sum_{i}a_if_*(E_i)=K_X+\sum_{i}a_if_*(E_i)$. My question is what is $\sum_{i}a_if_*(E_i)$ in Pic(X)? Is it zero?

For example X is smooth and Y is blowup one point in X. Then what is $f_*{E}$ as a sheaf on X? As Remy pointed out, it is may not be a line bundle. Is there a general theorem about this issue?

Shuhang
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  • It is not in general true that the pushforward of a line bundle is a line bundle. I think I know how to prove that it is true for a blowup, but it's a bit cumbersome. Alternatively, if $X$ and $Y$ are locally factorial (e.g. smooth), you can think in terms of Weil divisors, and use proper pushforward for cycles. In both cases you get that $f_* E$ is trivial. (If pushforwards of line bundles are line bundles, one then also has to check that these two definitions of $f_*$ agree; another painful task.) – Remy Nov 21 '15 at 17:36
  • Thanks for pointing out. $f_$ can either be Gysin map or push-forward of sheaves. In the case X and Y are smooth of same dimension, the two would be same. But now let's consider X has a singularity and Y is a resolution, and $f_$ is push-forward sheaf. – Shuhang Nov 21 '15 at 19:15
  • Could you please state what you mean by a resolution? There are many (slightly) different definitions for this. – Remy Nov 21 '15 at 19:32
  • For example $p\in X$ is a singular point. $f:Y\longrightarrow X$ is a resolution(at p), if Y is smooth in a neighborhood of $f^{-1}(p)$ and the restriction $f: Y-f^{-1}(p) \longrightarrow X-{p}$ is isomorphic. – Shuhang Nov 21 '15 at 20:05

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