7
votes
5answers
550 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 ...
3
votes
2answers
426 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 ...
7
votes
4answers
817 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 ...
0
votes
2answers
217 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 ...
2
votes
3answers
330 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 ...
6
votes
6answers
2k 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 ...
17
votes
5answers
2k 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 ...
4
votes
3answers
944 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 ...
11
votes
7answers
2k 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 ...