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I've been trying to get some intuition for flat morphisms. One fact I'm familiar with that has helped me understand this common phrase "flatness is the algebraic notion corresponding to fibers varying nicely" is the following. "If $X\rightarrow Y $ is a flat morphism between varieties then it's open and the fibers have constant dimension".

If I have a map of varieties $X\rightarrow Y$ that is open with fibers of constant dimension then is there another condition I could impose on the map to guarantee flatness? Ideally condition which can be explained geometrically.

Dcoles
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  • Probable duplicate of this – KReiser Nov 25 '21 at 22:51
  • @KReiser It is not a duplicate of the linked post. – Dcoles Nov 25 '21 at 23:32
  • Did you read it? $X$ Cohen-Macaulay and $Y$ regular implies the result you're after, and this is the top-voted answer there. – KReiser Nov 25 '21 at 23:55
  • Yes I did. I asked for a condition on the maps. Not a condition on the varieties. – Dcoles Nov 27 '21 at 03:11
  • Ah, it was not clear to me that you were asking for a condition on the map. Perhaps it might be a good idea to edit that in to your post so you get the sort of answers you want. (Separately, I'm not sure there are a lot of conditions that are just on the morphism - basically any flatness criteria I can think of off the top of my head involve some sort of condition on at least one of $X$ and $Y$. Good luck, though!) – KReiser Nov 27 '21 at 20:40

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