3
votes
1answer
37 views

Isomorphism of Grassmannians

I want to prove that two CW complexes $\mathrm{Gr}_{n}(\mathbb{R}^{n+k})$ and $\mathrm{Gr}_{k}(\mathbb{R}^{n+k})$ are isomorphic to one another. I'm pretty sure I can just show that the number of ...
2
votes
1answer
63 views

Affine variety example.

I know how to show the statement. But I cannot find an example (the part I underlined by a yellow pen) please help me for finding an example. Note: For example, can I consider the following ...
13
votes
5answers
382 views

Introduction to ring theory?

I've been teaching myself algebra these couple of months. I already went through the basics of group (Lagrange, action, class equation, Cauchy and Sylow theorems etc.) And I already have some linear ...
4
votes
1answer
165 views

Learning Complex Geometry - Textbook Recommendation Request

I wish to learn Complex Geometry and am aware of the following books : Huybretchs, Voisin, Griffths-Harris, R O Wells, Demailly. But I am not sure which one or two to choose. I am interested in ...
2
votes
0answers
34 views

Why can the group of isomorphism classes of line bundles be identified with $H^1(C,\mathbb O_C^*)$?

This is a reference request to the fact in the title. Is there a book at most as advanced as Hartshorn's which explains this result?
6
votes
1answer
149 views

How much Differential Geometry is needed to appreciate Algebraic Geometry?

I want to start self-studying Algebraic Geometry at some point in the near future. There are plenty of posts discussing prerequisites, but one thing I couldn't find a discussion of: How much ...
2
votes
1answer
127 views

Non-isomorphic $\mathbb{C}$-algebras

The question is as follows: Show that the $\mathbb{C}$-algebras: $A=\mathbb{C}[x,y]/(x^2y-xy)$, $B=\mathbb{C}[x,y]/(x^2y+xy^2)$, $C=\mathbb{C}[x,y,z]/(xy, yz, zx)$, and ...
10
votes
3answers
432 views

What is a good book to study classical projective geometry for the reader familiar with algebraic geometry?

The more I study algebraic geometry, the more I realize how I should have studied projective geometry in depth before. Not that I don't understand projective space (on the contrary, I am well versed ...
0
votes
1answer
52 views

Properties of Quasi-Coherent Modules

Let $X=\mathrm{Spec}\,A$ be an affine scheme and $M$ an $A$-module. Show that the following two conditions are equivalent: (a) $\tilde{M}$ is a locally free $\mathcal{O}_{X}$-module of finite type. ...
2
votes
1answer
57 views

Scheme-Theoretic Nakayama's Lemma

Let $X$ be a noetherian scheme and $\mathscr{F}$ a coherent $\mathscr{O}_{X}$-module. For a point $x \in X$, let $k(x)=\mathscr{O}_{X,x}/\mathfrak{m}_{x}$ be the residue field at $x$. (a) Suppose $x ...
31
votes
2answers
2k views

Why learning modern algebraic geometry is so complicated?

Many students - myself included - have a lot of problems in learning scheme theory. I don't think that the obstacle is the extreme abstraction of the subject, on the contrary, this is really the ...
2
votes
2answers
348 views

Books for algebraic geometry, algebraic topology [duplicate]

Possible Duplicate: (undergraduate) Algebraic Geometry Textbook Recomendations I am planning to self-study one of these two subjects: Algebraic geometry , Algebraic topology. I can borrow ...
11
votes
2answers
2k views

Path to Basics in Algebraic Geometry from HS Algebra and Calculus?

In this question, Why study Algebraic Geometry?, Javier Álvarez, develops a succint but encompassing description of algebraic geometry and its spread across different areas of mathematics. Indeed, it ...
6
votes
1answer
227 views

Basics of schemes and morphisms of schemes

I'm currently reading through Hartshorne, and have come across a few things that have left me wondering. (i) Somewhat pedantic, but also because I don't actually know the answer, (in Example 2.3.4) ...
7
votes
2answers
572 views

What is required to learn about algebraic geometry?

I want to learn about classical algebraic geometry. So what are subjects that are required to start learning about it? (Some preknowledge of algebra, commutative algebra?)
5
votes
2answers
403 views

Self-Teaching: Is Geometry the Nexus of all Mathematics?

Necessary prologue: I'd really like to become more fluent in the language of mathematics. I don't have a schedule that permits me taking a class and any on-line tutors that I find seem relatively ...
2
votes
1answer
968 views

Can I go through Hartshorne without knowing much analysis?

I know intro abstract algebra and some real analysis. Is this enough to study algebraic geometry from the book of Hartshorne?
12
votes
3answers
1k views

Learning schemes

Could someone suggest me how to learn some basic theory of schemes? I have two books from algebraic geometry, namely "Diophantine Geometry" from Hindry and Silverman and "Algebraic geometry and ...