9
votes
1answer
139 views

Modern Research in Algebraic Topology

What are some of the main directions and trends in modern (let's say within the last ~10 years) algebraic topology? What are some major open problems or recent results? In a more specific direction, ...
8
votes
0answers
71 views

Why is the Leibniz rule a sufficient ingredient in the construction of the tangent space?

This is a very soft question, but I am wondering if anyone can shed light on why it is that the product rule (and linearity) provide exactly the right requirement for the space of derivations to be ...
2
votes
2answers
84 views

What is the use of scheme theory?

I should preface this by saying that my background in Algebraic Geometry is (more or less) the content of Vakil's notes up through Chapter 4 (i.e. through the definition of a scheme and several ...
5
votes
1answer
78 views

Thinking About Fractional Ideals Geometrically

So algebraic geometry gives one a way of thinking about about rings geometrically. Like prime ideals correspond to points in the spectrum of a ring, maximal ideals are closed points and so on. This ...
0
votes
0answers
41 views

Is it important to study plane algebraic curves before read Fulton's book

I'm studying Fulton's algebraic curves book and I would like to know how important study plane algebraic geometry before read Fulton's book. Example of books on this subject: Algebraic Curves - ...
7
votes
3answers
338 views

Video lectures of algebraic geometry (Hartshorne, Shafarevich, … )

I am a commutative algebra student. I wonder if there is some video lectures of algebraic geometry courses available online for free? I'd like the lectures to cover main topics of the books ...
0
votes
0answers
62 views

Connection between differential form of a manifold and Sheaf of relative differential of a map of schemes.

I was wondering whether there is a connection between the differential form of a manifold and Sheaf of Relative differential of a Scheme map. Definitions: Differential form on a manifold M is a map ...
3
votes
0answers
117 views

Abstract algebraic geometry vs complex algebraic geometry

Sorry in advance if my question is not precise enough. I'm currently trying to study algebraic geometry on my own. I've started by trying to read Harsthorne and Liu's book. And i found it very ...
3
votes
0answers
82 views

Is there any visual animation to show the basic concept of algebraic geometry? [closed]

Is there any visual animation to show the basic concept of algebraic geometry? There are rarely pictures in textbooks, so are there any animation to show basic but important concepts?
0
votes
0answers
60 views

Difference between isomorphisms, deformations, diffeomorphisms and homeomorphis,s

I am learning about complex algebraic varieties and have encountered several equivalence relations: isomorphism, birational, deformation equivalent, diffeomorphic and homeomorphic. I consider all of ...
18
votes
3answers
338 views

Why there is much interest in the study of $\operatorname{Gal}\left(\overline{\mathbb Q}/\mathbb Q\right)$?

Let's start for a simple quote from wikipedia: "No direct description is known for the absolute Galois group of the rational numbers. In this case, it follows from Belyi's theorem that the ...
2
votes
0answers
50 views

Idea behind the concept of schemes

Having taken an introductory course on algebraic geometry (without introducing schemes), the notion of schemes seems to be quite unrelated to all we've done there. What are the most important reasons ...
2
votes
1answer
125 views

What should a student (with algebraic-geometry minded) study in differential geometry?

One of my friend who is an undergraduate student, has known something about algebraic geometry (equivalent to chapter 1 and a little bit chapter 2 in GTM 52 by Hartshorne). He is now has to study a ...
5
votes
2answers
306 views

So can anybody indicate whether it is worthwhile trying to understand what Mochizuki did?

So I am looking at some math stuff and I start looking at the abc-conjecture. Naturally I run into the name Mochizuki and so start trying to see what he did. Well, he is starting look like another ...
3
votes
3answers
75 views

How to tell if algebraic set is a variety?

I've been reading some basic classical algebraic geometry, and some authors choose to define the more general algebraic sets as the locus of points in affine/projective space satisfying a finite ...
3
votes
0answers
61 views

Good confusion-avoiding notation for gluing toric varieties from fans?

$ \newcommand\R{\mathbb R} \newcommand\C{\mathbb C} \DeclareMathOperator\Cone{Cone} $I'm trying to establish a good notation to avoid confusion when we glue toric varieties from affine pieces. A ...
7
votes
2answers
1k views

Algebraic geometry project ideas for high school students

I am teaching a "senior seminar" course for strong students at our local high school. For 6 weeks the students learned about basic/classical algebraic geometry. In a few weeks they will start projects ...
11
votes
4answers
238 views

Topics in Differential geometry $\cap$ Algebraic geometry

I find (both: differential and algebraic) geometry fascinating. I'm just beginning my graduate studies, but I'd like to know some topics/theorems in the intersection of these two (since I don't really ...
4
votes
4answers
118 views

Why are roots of polynomials called geometric objects?

I read the following from the Wikipedia article about algebraic varieties: Proven around the year 1800, the fundamental theorem of algebra establishes a link between algebra and geometry by ...
8
votes
1answer
116 views

Proper VS. Projective morphism

Despite the obvious differences in definitions, I have a very strong impression that any theorem whose assumption requiring the morphism to be projective can be replaced by the morphism to be proper. ...
2
votes
1answer
37 views

Why do we want to define a $k$-scheme to be birational if the rational map (and its inverse) to $\Bbb A_k^n$ is over $k$?

Two varieties $X,Y$ are said to be birational if there exist rational maps in each direction such that either composition is the identity on a open dense subset. Note that here the morphisms aren't ...
2
votes
0answers
326 views

Is there any English version of Récoltes et Semailles?

I felt like my question isn't appropriate for MO, so I though maybe I should post it here. I want to read Alexander Grothendieck's "Récoltes et Semailles", but I don't know any French. I can easily ...
4
votes
2answers
108 views

What is a Kählerian variety?

I know what a Kähler manifold is, and I (roughly) know what a variety is. However, I don't know what a Kählerian variety is. Is it just a variety which is also a Kähler manifold, or is it a separate ...
6
votes
2answers
122 views

In algebraic geometry, why do we use $\mathbb C$ instead of the algebraic closure of $\mathbb Q$?

In algebraic geometry, why do we use $\mathbb C$ instead of the algebraic closure of $\mathbb Q$? What properties of algebraic varieties use the topological completeness of our field? I'd be ...
0
votes
0answers
64 views

Nonsingular curve

I am kind of curious about the nice properties between affine nonsingular curve and projective nonsingular curve. In my feeling to define sheaf of nonsigluar curve, most of the resources are focusing ...
4
votes
0answers
102 views

Generic Points to the Italians

When I first learned algebraic geometry, I naturally wiki-ed the subject and there was a line there that said the old school Italians used the notion "generic points without any precise definition." ...
10
votes
3answers
877 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 ...
12
votes
2answers
191 views

Are there applications of noncommutative geometry to number theory?

The marriage of algebraic geometry and number theory was celebrated in the twentieth century by the school of Grothendieck. As a consequence, number theory has been completely transformed. On the ...
6
votes
1answer
372 views

Why can algebraic geometry be applied into theoretical physics?

It is to be said at the outset that I do not have much familiarity with physics beyond what is in a semi-popular book; say, the Feynman Lectures Vol 1 and 2. As I progressed in math graduate school ...
4
votes
1answer
143 views

Motivation and Applications for Toric Varieties

I'm a graduate student of mathematics starting to study algebraic geometry with a focus on toric varieties (along Cox, Little, Schenk). From what I learned so far, I can grasp that toric varieties ...
4
votes
0answers
71 views

The importance of generating series

What follows is a very nebulous question. I just seek for some help in understanding a "technique" which has proven itself very powerful. I noticed that very often generating series appear in ...
13
votes
5answers
586 views

Favourite applications of the Nakayama Lemma

Inspired by a recent question on the nilradical of an absolutely flat ring, what are some of your favourite applications of the Nakayama Lemma? It would be good if you outlined a proof for the result ...
38
votes
3answers
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 ...
3
votes
0answers
140 views

What analysis is needed for AG?

My question: On what level do I need to know (complex/real) analysis or diff. geometry to study algebraic geometry from Hartshorne? And AG in general? Context: I will be taken a course in ...
3
votes
2answers
415 views

Local vs. global in the definition of a sheaf

Apologies in advance that this question is inescapably soft. What I am stuck on is squishy; I have the feeling that if I could even make it precise, I'd already be satisfied. To what extent is a ...
7
votes
2answers
487 views

Connection between algebraic geometry and high school geometry.

if there is one thing that going to math competitions has taught me it is that I suck at high school olympiad level geometry. However I often find solace in the fact that not a lot of mathematicians ...
10
votes
1answer
169 views

What are the “correct” modules over locally ringed spaces?

$$\begin{array}{ccccc} \text{schemes} & \longrightarrow & \text{locally ringed spaces} & \longrightarrow & \text{ringed spaces} \\ | && | && | \\ \text{quasi-coherent ...
4
votes
0answers
90 views

Is there any equivalence between the category of schemes over $\mathbb R$ and the category real manifolds

The equivalence of the category of smooth projective curves over $\mathbb C$ and the category of compact Riemann surfaces is, I believe, well documented. For example, it is mentioned on the wiki page: ...
2
votes
0answers
170 views

Algebraic curves and riemann surfaces

I am a physics undergrad with no formal background in complex analysis. I have done complex analysis at the level of the first 4 chapters (till Complex integration) from Churchill and Brown. I am very ...
4
votes
2answers
154 views

To what extent is a scheme morphism determined by its topological map?

I am just beginning to learn scheme theory. This question is aimed at getting a feel for something so apologies in advance for the lack of precision. I am struck by the following difference from the ...
31
votes
7answers
962 views

Appearance of Formal Derivative in Algebra

When studying polynomials, I know it is useful to introduce the concept of a formal derivative. For example, over a field, a polynomial has no repeated roots iff it and its formal derivative are ...
6
votes
1answer
352 views

Prerequisites for studying Hodge theory and the Hodge conjecture

To what branch of mathematics does the Hodge conjecture belong? I'm aware that it's very advanced, but what kind of prerequisites would one need to understand those problems? Can you suggest some good ...
13
votes
2answers
3k 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 ...
5
votes
1answer
410 views

Examples of algebraic techniques from algebraic geometry solving geometric problems.

Classical algebraic geometry begins by interpreting the solutions to polynomial equations as geometric objects. The solutions can then be studied geometrically, and a correspondence between their ...
72
votes
2answers
12k views

Why study Algebraic Geometry?

I'm going to start self-stydying algebraic geometry very soon. So, my question is why do mathematicians study algebraic geometry? What are the types of problems in which algebraic geometers are ...
9
votes
1answer
230 views

Good references for stacks

I have seen stacks come up in various settings recently. I understand that, at least heuristically, they are some sort of generalization of a scheme, but I don't actually know anything about them. ...
10
votes
1answer
167 views

Weil conjectures - motivation?

Can anyone explain (heuristically, intuitively is fine) what the importance of the Weil conjectures is? I realize they have motivated much of recent algebraic geometry. I don't really understand why ...
3
votes
1answer
285 views

Chapter V of Grothendieck's EGA

Grothedieck wrote, in the introduction of his EGA, Chapter V would be Procedes elementaires de construction de schemas(Elementary procedures for construction of schemes). I wonder what he meant by it. ...
2
votes
1answer
84 views

Why do number rings have no endomorphisms

This question is about the analogy between number fields and function fields. It's a soft question and the title misrepresents the question. Consider the projective line over a field. This has many ...
10
votes
2answers
472 views

Varieties as schemes

Some questions about schemes and varieties, one really basic. I follow the definitions as given in Hartshorne. Firstly, my main question. I understood that Grothendiecks introduction of schemes ...