Here is a place to post suggestions, topic requests, and in particular, papers (old or new) that fit the general feel of the blog (whatever that is) and would be good topics for discussion. We’ll see how this goes.

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Rigorous trivialities is a web log about mathematics, but especially geometry, broadly construed. Contributors will be Charles Siegel, Jim Stankewicz and occasionally Matt Deland. Charles specializes in algebraic geometry, topology and mathematical physics. Jim specializes in arithmetic algebraic geometry. Matt has transitioned from algebraic geometry to work in industry.

Header is taken from the larger work by fdecomite under the creative commons license.

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Just a note – I’m hosting the next Carnival of Mathematics at thoughtcounts.net tomorrow. The carnival page lists no host, so I have a lower than usual number of submissions. If you have anything to submit, I’d love to hear from you.

Hi, I’m an undergraduate student and I’m interested in AG. Now that I began to read some books I found very often the word “generic”. So I think it would be nice if someone post a blog about it. I found a definition (Griffiths-Harris book) of the generic word in AG but it’s too technical.

Well, I can give you it roughly here (a longer post will follow when I can find the time). In AG, open sets are always dense (well, in irreducible things). So generic will generally mean that something holds for an open set on the space of all such objects, and occasionally it will mean the countable intersection of open sets. Think that it means that away from some variety (ie, there are some polynomial equations saying when it fails) something is ok, though you might have to deal with a variety with countably many irreducible components.

Thank you so much for your help. I hope you find the time soon. I’ll be waiting for it.

I tend to think of “generic” as meaning “it stays the same when you wiggle it”.

Two curves being tangent is not a generic intersection. Wiggle them a bit and a crossing tangency becomes a simple crossing, or a noncrossing tangency either becomes a pair of crossings or vanishes entirely.

On the other hand, two curves crossing is generic, since if you wiggle them a bit they’ll still cross.

Hi Charles,

I’ve looked on your page at the oral exams: can you please tell me what do mean?

Also, after how many years (if any) of grad school did you take those oral exams?

Thanks

I took my oral exams right at the beginning of my second year. And here’s a quick answer, a on a curve is a linear system of dimension and degree . So it maps the curve to a degree curve in .

Hi, Charles,

What is “essentially of finite type morphism” (from Koll’ar’s book)? If a scheme is “essentially of finite type” over a locally noetherian scheme, does it imply, that it is also locally noetherian? Sorry for presumably trivial question.

For rings, we say that is essentially of finite type as an -algebra if it is a localization of a finitely generated -algebra. So presumably, for schemes, it would mean that this holds on affines, and perhaps require that the map is affine to make it work out, haven’t thought it through very deeply.

Now, this says that if is noetherian, then is, so my thought is that a scheme, essentially of finite type over a locally noetherian scheme should also be locally noetherian.

Have you collected all your “AG from the beginning” posts somewhere?

I have tried to use some kind of search on this blog but the results didn’t satisfy me. So maybe you could just assemble an ordered list (or a pre-ordered list, a lattice) of those blog entries?

I would want to start kind of in-between but I don’t know where… and the old postings don’t link to the newer ones..

By the way: thanks for having already written so much useful stuff!

They’re in the category “AG from the Beginning” but I guess I’ll make a page, and then whenever I do something somewhat basic, I’ll link to it there.

Hi,

I have a question about proper maps: if f: Y −→ Z is a proper beetween quasi projective varieties and let

X ⊂ Y such that the restriction of f to X is again proper is the embedding of X into Y proper?

Thank you

@Matteo: An immersion is proper iff it is closed.

Just wanted to drop a note to say I’ve been finding your AG from the beginning blog posts extremely helpful in my own studies, thank you for such a readable introduction to AG!

Hello Charles,

I want to thank you for the excellent work you have done through your blog. It has made my life a lot more comfortable tackling the demon that is Algebraic geometry.

I would really love it if you could write a similar blog for Commutative algebra, as it is a subject that is quite often ignored a thorough treatment when instructors teach algebraic geometry.

Well, I am preparing to return to blogging (watch for a post in about a week) though my plan was a bit more analytic in nature. I’m more on the analytic side of algebraic geometry, that draws from complex analysis, differential geometry and algebraic topology as the main tools. However, and I know this is a comment so it won’t be noticed as much, if other people are interested in some sort of “commutative algebra from the beginning” series, trying to be informal and talk about the intuition behind commutative algebra constructions and theorems and connecting them to algebraic geometry, reply here, and I might do it.

“trying to be informal and talk about the intuition behind commutative algebra constructions and theorems and connecting them to algebraic geometry”

That sounds absolutely perfect, if you have the time. Your AG from the Beginning series is great by the way!

comment 3 on this page :https://rigtriv.wordpress.com/2009/11/03/chern-classes-part-1/ is spam and not something authorised by the linked company. appreciate removal help.

Removed, I haven’t been maintaining the blog for awhile, so the comment hygiene has declined. Apologies.

Hi, I want to start doing some serious studies of Algebraic Geometry. Usually, the way I study math is to understand the theorems, definitions and corollaries the best I can and then do a lot of exercises. However, I don’t know if you could recommend any textbook of AG that has solutions so that I can do exercises and then check myself with the solutions.

Thanks!

Any new posts coming up? Just found you guys and hoping to see more!

Could you please do a post on Grothendieck Fundamental Group?

Thank you.

Hello! I had a question about Japanese actually–I came across the word 原始ベクトル in a paper on tropical geometry, but I’m not sure what it means. Do you have any idea? A “primitive” vector? Thank you!