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I am reading the lecture note of Dori Bejleri about Picard schemes: https://people.math.harvard.edu/~bejleri/teaching/math259xfa19/math259x_lecture12.pdf

In Example 12.8, I don't understand why the smooth and irreducible generic fiber implies that $f:X\rightarrow S$ is a universal algebraic fiber space, i.e. $f_* \mathcal{O}_X \xrightarrow{\sim} \mathcal{O}_S$ holds universally.

I have tried to find some conditions to be a universal algebraic fiber space, but I only find a condition in the notes of Picard schemes of Kleiman (Exercise 3.11, when $f:X \rightarrow S$ is proper and flat and its geometric fibers are reduced and connected): https://arxiv.org/abs/math/0504020

Can anyone help me to explain the example of Bejleri? Are there any other conditions implying this universal isomorphism?

hungnk1998
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  • If you mean Example 1 in the linked PDF, I definitely should have added an assumption that all the fibers are reduced. The example is local on the base so you can for example shrink $\mathbb{A}^1$ to some open neighborhood of $0$ which avoids non-reduced fibers or you can pick $f$ appropriately. – Dori Bejleri Oct 10 '21 at 03:52
  • Thank you very much. – hungnk1998 Oct 11 '21 at 13:48

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