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In Hartshorne's Algebraic Geometry Exercise I.4.8,

Exercise: Show that any variety of positive dimension over $k $ has the same cardinality as $ k $.

Hints: Do $\mathbb{A}^{n} $ and $\mathbb{P}^{n}$ first. Then for any $ X $, use induction on the dimension $ n $. Use (4.9) to make $ X $ birational to a hypersurface $ H \subset\mathbb{P}^{n+1 }$. Use (Ex. 3.7) to show that the projection of $ H$ to $\mathbb{P}^{n}$ from a point not on $ H$ is finite-to-one and surjective.

Although this has been asked in many posts in MSE, namely:Cardinality of variety and Cardinality of quasiaffine variety . My question is different. In these posts, it seems that the proof provided (although they are indeed wonderful and helpful) did not follow the hints by Hartshorne. I tried to follow the hints, but didn't get the promising results.

So my question is: How to finish this exercise using the methods provided in the HINTS?

My attempts:

(1) I had made some attempts and found that using Noether normalisation theorem, we may direct prove the desired result. Maybe I will post an answer using this in the posts mentioned above. Yet I still hope to know how to follow the hint.

(2) I have managed to prove that the cardinality of $\mathbb{A}^{n} $ and $\mathbb{P}^{n}$ are the same as the cardinality of the ground algebraically closed field $k$. Yet what's next?

Hetong Xu
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  • https://en.wikipedia.org/wiki/Tarski%27s_theorem_about_choice – Kenta S Jul 29 '20 at 07:36
  • It's rare for perfect texts to be published, and Hartshorne's no exception - there are occasional (or more than occasional, depending on who you ask) lemons and hints that aren't quite as useful as they could be. Following the hint you'd make $X\sim H$, then project $H$ to $\Bbb P^n$, and show the image of the open isomorphic to an open in $X$ contains a nontrivial open subset of $\Bbb P^n$, and then if you can show that a nontrivial open subset of $\Bbb P^n$ has cardinality $|k|$, you're done. But there's no need to prove that last bit by induction - and many solutions don't. – KReiser Jul 29 '20 at 09:10
  • All this is to say that I sympathize with your plight of not understanding what Hartshorne's going for in the hint, and that's the best I could think of under the circumstances. Perhaps someone else will have a better idea what Hartshorne's intent was. – KReiser Jul 29 '20 at 09:14
  • @KReiser Thank you for your comments! Yet could you explain more on the line "......and show the image of the open isomorphic to an open in $X$ contains a nontrivial open subset of $\mathbb{P}^n$". I don't know how to understand this sentence. (So sorry since English is a foreign language for me.) – Hetong Xu Jul 29 '20 at 13:40
  • $X$ birational to $H$ means that there's open subsets $U\subset X$ and $W\subset H$ so that $U\cong W$ - in particular, this is a bijection. Next, letting $\pi: H\to \Bbb P^n$ be the projection, we want to show that $\pi(W)$ contains a nonempty open set in $\Bbb P^n$. – KReiser Jul 29 '20 at 17:12
  • @KReiser Thank you so much for your comments! – Hetong Xu Jul 30 '20 at 01:34
  • @KentaS Sorry but I viewed the wikipedia page, do you mean that we can use the following Tarski's theorem about choice: For every infinite set $A$, there is a bijective map between the sets $A$ and $A \times A$. And "Trichotomy: If two sets are given, then either they have the same cardinality, or one has a smaller cardinality than the other. Given two non-empty sets, one has a surjection to the other." But I cannot see the relation of this theorem to my question. Could you clarify it? Thank you :) – Hetong Xu Jul 30 '20 at 01:39

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