Timeline for answer to Zariski connectedness theorem: from analytic & topological viewpoint by SeanC
Current License: CC BY-SA 4.0
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| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 25, 2024 at 10:59 | history | bounty awarded | user267839 | ||
| Dec 25, 2024 at 10:59 | vote | accept | user267839 | ||
| Dec 25, 2024 at 4:03 | comment | added | SeanC | Your guess is right. However, case 2 also requires smoothness to make "irreducible = connected". If $f$ is only assumed flat, then case 1 still works with "irreducible" replaced by "reduced and irreducible", while case 2 still works with "irreducible" replaced by "connected", each with the same argument. However, the locus of $s \in S$ such that $f^{-1}(s)$ is geometrically irreducible is neither open nor closed in general without assuming smoothness of $f$. | |
| Dec 25, 2024 at 1:36 | comment | added | user267839 | In last two lines: why smoothness of $X^1$ & emptyness of its generic fibre implies that $X^1$ is itself empty? A guess: There, when you write that "$X^1$ is smooth", you still mean tacitly as "relative smooth" over $S=\operatorname{Spec}(B)$ , right? And so especially flat,and then the fibre dimension stays constant. So it seems - if I understood your reasoning there correctly - that emptyness of $X^1$ only depends on demanded flatness of $X^1 \to S$ alone, right? Or did you there had in mind a different argument to conclude that $X^1$ must be empty? | |
| Dec 24, 2024 at 22:53 | comment | added | LSpice | Re, ah, thanks! | |
| Dec 24, 2024 at 22:28 | comment | added | SeanC | There are five published chapters of EGA: 0, I, II, III, and IV. Chapters I and II each take up one volume; III takes up two; and IV takes up four. Chapter 0, on the other hand, starts before Chapter I and is then augmented when necessary for the others. Chapter 0_III refers to the part of Chapter 0 that appears in (the first volume of) Chapter III. | |
| Dec 24, 2024 at 22:06 | comment | added | LSpice | What is EGA 0_III? | |
| Dec 24, 2024 at 17:16 | history | answered | SeanC | CC BY-SA 4.0 |