The Geometry of Schemes [pdf] | Hacker News Reader

The Geometry of Schemes [pdf]

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Why is this on the front page of hacker news? Hopefully that comes across as a genuine question and not snark. I mean as an ex-mathematician I'm thrilled, but schemes are an incredibly abstract object used in an incredibly abstract branch of mathematics (algebraic geometry).

I have the same question in my mind. Also thrilled though. I think there is a genuine fascination in HN and in general with Grothendieck [1], [2].

[1]: https://en.wikipedia.org/wiki/Alexander_Grothendieck [2]: https://hn.algolia.com/?q=Grothendieck

Interesting, yeah. I guess he was the mathematical equivalent of the "rogue" archetype. Brilliant, did things in his own way, total lack of respect for authority, shrouded in mystery. I can definitely see the appeal =)

Sometimes, the best way to learn about abstruse topics one has a passing curiosity in is to upvote what pops up on HN and hope that some nerd might drop by and comment with a simplified intuitive picture for plebs :-)

(Edited to be more helpful)

These days, some nerds prefer to ask AI to confirm their "precious" intuitions of why schemes might be needed in the first place. To fix the problems with certain basic geometric notions of old timers? They are then so spooked that the AI instantly validates those intuitions without any relevant citations whatsoever that they decide not to comment

But still leave warnings to gung-ho nerds in the form of low-code exercises

That's a theory, but I think it's more likely that the few people in the world who deeply understand schemes are locked in the basement of a mathematics department somewhere, and not on hacker news =P

Ah those delvers who remember their lock combinations might still visit HN then (hello @Syzygies?

https://raw.githubusercontent.com/Syzygies/log_folders/maste...

And ahem ahem singularities https://johncarlosbaez.wordpress.com/2025/11/05/the-inverse-... )

> That's a theory, but I think it's more likely that the few people in the world who deeply understand schemes are locked in the basement of a mathematics department somewhere, and not on hacker news =P

I rather think that because of the very low career prospects in research, quite a lot of people who are good in this area rather left research and took some job in finance or at some Silicon Valley company, and thus might actually at least sometimes have a look at what happens on Hacker News. :-)

Lean4 is "sponsored" by Microsoft

See pages 9-10, 14 https://web.archive.org/web/20250818144653/https://aitp-conf...

Also https://youtu.be/g2--VL2SkMo?t=44m42s

I think you overestimate how many people exist in the world with a professional interest in algebraic geometry! The vast majority of mathematicians have no idea how to compute with schemes (and there aren't that many of them to begin with).

Even though I am from in a different area of mathematics, I know quite many people who work(ed) in algebraic geometry (and at the university where I graduated there wasn't even an academic chair for (Grothendieck-style) algebraic geometry).

The amount of people I know who would love to learn this material is even many, many magnitudes larger (just to give some arbitrary example: some pretty smart person who studied physics, but (for some reasons) neither had any career prospects in research nor found any fullfilling job, who just out of boredom decided that he would love to get deeply into Grothendieck-style algebraic geometry).

It's hard to help GP but I'm gonna try (pls forgive me):

I believe that the masses don't have a deep understanding of Schemes because of enemy action by the sufficiently advanced stupidity (aka loneliness) of the intelligent :)

Some (ex-)academics get triggered by the so-thought foolhardy Buzzard & his undergrads, but B+Co are, at least, not being that kind of imbecile https://spiral.imperial.ac.uk/server/api/core/bitstreams/012...

https://github.com/ramonfmir/lean-scheme

Their interest is "pro" and they are not a hypothesis

(& I'd NOT bet against that they understand deeper than Sturmfels and his students)

Schemes (like cat theory) have become a sort of religion-- it's sad because Grothendieck himself might not have understood them intuitively.. and it won't be the first time.. Feynman didn't understand Path Integrals, nor Archimedes integration!! BECAUSE they were all loners whose first resort was WRITING LETTERS

Ps: as with Jobs.. I hesitate to call Buzzard a full-time salesman

If you want to hang out in meatspace: do you have a public key?

I guess we hang out in different academic circles. I met a single algebraic geometer in my whole academic career. But people are into very different stuff where I come from, which may have biased me (topology, number theory and category theory for the most part, and a lot of relativity/fluid dynamics on the applied side of the department). Based on rough estimates from papers published on arxiv over the last few years, I (very) conservatively estimate there are ~5000 working algebraic geometers in the world right now.

> The amount of people I know who would love to learn this material [...]

I am one of them =) but my point wasn't really about people who want to learn the material (which I assume includes many orders of magnitude more humans) it was about people who already deeply understand it.

in many educational systems, aptitude in math (the more abstract, the better) is conflated with intelligence. so maybe many of us have internalized we should valorize it?

Teach biologists that?

https://web.archive.org/web/20250115224532/http://www.dam.br...

https://golem.ph.utexas.edu/category/2014/12/can_one_explain...

https://liorpachter.wordpress.com/2014/12/30/the-two-culture...

(2014)

With someone actually trying in 2002, using TFB for inspiration & M2 as a very early Jupyter/sage

https://web.archive.org/web/20250000000000*/https://mast.que...

(These bloggers as well as creator of M1 may want to chip in a dozen cts)

What's up with the raccoon on page 67?

oh yeah! that is insane.

I think it's cute!

What if it's steganography or symbolic or a puzzle.

If it's none of that, to complain about it is a bit too harsh.. for something on page 67.

Pretty sure that's a ferret.

I know both authors, so I wrote them to ask. This is a rogue image that appears in various PDFs posted online, but not on Springer's official PDF or any print copy we've seen.

Published in 2000.

(I studied schemes 10 years before, but I quit maths in 2000 so this book wouldn't have helped me. It seems like a good introduction, looking at the TOC. Grounded on actual geometry, not just category theory like other textbooks).

Also, the racoon ?!

The raccoon has ruined this for me…

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