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5 votes
0 answers
2k views

A course on modern algebraic geometry from "The Stacks Project"

I hope this question is viable for this site. I'm sincerely sorry, if you think it isn't. For a lot of time, "EGA" by Alexander Grothendieck and Jean Dieudonne was "the" reference on the basics of ...
TavukKaghul's user avatar
13 votes
2 answers
2k views

teaching higher algebra

Has anyone ever (successfully or unsuccessfully) taught a course in higher algebra (in the $\infty$-categorical sense)? I'm asking out of curiosity (and also hoping for more resources). The kind of ...
pro's user avatar
  • 534
13 votes
1 answer
605 views

A funny factorization of the Jacobian coming from the lines on the Fermat cubic

Here is something which came up in my algebraic geometry class, and I'm wondering if it has a deeper explanation. Let $F(w,x,y,z) = w^3+x^3+y^3+z^3$ and let $X$ be the cubic surface in $\mathbb{P}^3$ ...
David E Speyer's user avatar
11 votes
0 answers
2k views

Total spaces of tangent/cotangent bundles in a course where all varieties are quasi-projective

$\def\PP{\mathbb{P}}$In a course where all varieties are quasi-projective (as in Shafarevich Volume I), I am trying to figure out whether I can justify talking about the total spaces of the tangent ...
David E Speyer's user avatar
8 votes
0 answers
554 views

Lower semicontinuity of naive fiber size

I would like to present the following result in my algebraic geometry class, but it is seeming much harder than I would expect. Since my class is working with closed points over an algebraically ...
David E Speyer's user avatar
27 votes
8 answers
5k views

Conceptual algebraic proof that Grassmannian is closed in Plücker embedding

I'm planning lectures for my intro algebraic geometry course, and I noted something awkward that is coming up. We're starting projective varieties soon. Of course, we'll prove that projective maps are ...
David E Speyer's user avatar
7 votes
4 answers
841 views

Easy to state applications of dimension theory in algebraic geometry

Dimension theory is quite a sophisticated topic (at least for me), it is fully settled in Shafarevich's book on the first 100 pages. Shafarevich gives two nice applications of the theory. 1) A proof ...
aglearner's user avatar
  • 14.3k
7 votes
3 answers
1k views

Higher dimensional Bezout via Hilbert polynomials: a reference

For the purposes of teaching my elementary course in algebraic geometry I am looking for a reference (or notes) that contains a complete proof of a higher-dimensional weak Bezout theorem. I only want ...
aglearner's user avatar
  • 14.3k
14 votes
5 answers
2k views

"Classical" consequences of Bezout's theorem in dimensions $>2$

By Classical I mean something that could have been found before 1900 (say). A well known consequence of Bezout's theorem for plane curves is Pascal's theorem http://en.wikipedia.org/wiki/Pascal'...
aglearner's user avatar
  • 14.3k
0 votes
2 answers
562 views

Lines on degree 2n-3 Fermat hypersufaces

It is well known that a generic hypersurface of degree $2n-3$ in $\mathbb CP^n$ has finite number of lines. I would like to ask a couple of questions about lines on Fermat hypersurfaces and their ...
aglearner's user avatar
  • 14.3k
23 votes
12 answers
15k views

Textbook for undergraduate course in geometry

I've been assigned to teach our undergraduate course in geometry next semester. This course originally was intended for future high-school teachers and focused on axiomatic, Euclid-style geometry (...
2 votes
3 answers
410 views

Pedagogical notes on line bundles on complex projective manifolds

I would like to find some notes (or book), that explains on a very basic level what is a line bundle on a complex projective manifold. Maybe even, what is a line bundle on $\mathbb CP^n$. It seems ...
17 votes
6 answers
7k views

Explaining the concept of projective space: notes for students

This is a question on teaching. I am teaching at this moment a course in algebraic geometry for master students on a very basic level. Today (this was the fourth lecture) I discovered that only four ...
27 votes
5 answers
5k views

Varieties as an introduction to algebraic geometry / How do professional algebraic geometers think about varieties

This really is two questions, but they are kind of related so I would like to ask them at the same time. Question 1: In a question asked by Amitesh Datta, BCnrd commented that it is important to ...
7 votes
3 answers
3k views

The etale fundamental group of a field

Background and motivation: I am teaching the "covering space" section in an introductory algebraic topology course. I thought that, in the last five minutes of my last lecture, I might briefly sketch ...
Charles Staats's user avatar
21 votes
7 answers
3k views

What should be taught in a 1st course on Riemann Surfaces?

I am teaching a topics course on Riemann Surfaces/Algebraic Curves next term. The course is aimed at 1st and 2nd year US graduate students who have have taken basic coursework in algebra and manifold ...
jlk's user avatar
  • 3,284
18 votes
12 answers
10k views

Looking for an introductory textbook on algebraic geometry for an undergraduate lecture course

I am now supposed to organize a tiny lecture course on algebraic geometry for undergraduate students who have an interest in this subject. I wonder whether there are some basic algebraic geometry ...