3
votes
3answers
57 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
43 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
568 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
187 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
106 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
62 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
33 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
105 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
103 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 ...
10
votes
0answers
167 views

Important topics for arithmetic geometry (esp Arakelov geometry)?

Seeing other past successful 'roadmap' questions, I hope this question is acceptable and not too vague. I know I'd like to eventually study arithmetic algebraic geometry - but I also know that it's a ...
6
votes
2answers
112 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
58 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 ...
3
votes
0answers
88 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
429 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
148 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
233 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
106 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
67 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 ...
12
votes
5answers
462 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 ...
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 ...
3
votes
0answers
125 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 ...
4
votes
2answers
253 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
409 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
160 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
84 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
124 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
128 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 ...
30
votes
7answers
874 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
295 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 ...
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 ...
5
votes
1answer
357 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 ...
61
votes
2answers
9k 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 ...
8
votes
1answer
182 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. ...
9
votes
1answer
156 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
256 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
79 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 ...
9
votes
2answers
392 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 ...
12
votes
6answers
813 views

Algebraic Geometry Text Recommendation

I need to learn about Algebraic Geometry (perhaps from in the context of finite fields) and am looking for a recommendation for a text. Now, I've already done a search and checked out what was ...
4
votes
1answer
57 views

Can different uniformizations of Riemann surfaces be related somehow

Let $X$ be a hyperbolic compact connected Riemann surface. Let $U\subset X$ be an open subset. Assume that $U\neq X$. We can uniformize $X$ by $\mathbf{H}$ directly to obtain it as a quotient of ...
2
votes
1answer
306 views

Hard problems in algebraic geometry

People very often say that algebraic geometry is a hard subject and has many challenging problems to solve. I believe the hodge conjecture is the one of the most difficult in the field and you, which ...
9
votes
2answers
421 views

Variety vs. Manifold

In the ambit of differential geometry the aim is to study smooth manifolds. Why the objects studied in algebraic geometry are called algebraic varieties and not for example algebraic manifolds? I am ...
9
votes
1answer
377 views

Stacks in arithmetic geometry [closed]

Stacks, of varying kinds, appear in algebraic geometry whenever we have moduli problems, most famously the stacks of (marked) curves. But these seem to be to be very geometric in motivation, so I was ...
1
vote
1answer
93 views

Is the study of algebraic curve is techniquely equal to the advanced division of analytic geometry, if not, what is the difference?

Is the study of algebraic curve is techniquely equal to the advanced division of analytic geometry, if not, what is the difference? And what is other branch of advanced analytic geometry called? in ...
3
votes
1answer
96 views

Varieties given by non-algebraic equations

In algebraic geometry one (mostly) studies varieties given by polynomial equations. Such equations define algebraic varieties and there are many "dictionaries" available. For example, the category ...
5
votes
0answers
131 views

Why are injective $\mathscr{O}$-modules flasque?

Let $X$ be a topological space, and let $\mathscr{O}$ be a sheaf of rings on $X$. It is easy to verify that the functor $\Gamma (U, -) : \textbf{Mod}(\mathscr{O}) \to \textbf{Ab}$ is representable, ...
3
votes
1answer
194 views

Intuition on the definition of “rational maps”

I'm studying some representation theory on $S_n$ and $GL(V)$ and tensor spaces, and have come across a lot of material involving rational representations. I'm not really an algebraic geometer by ...
3
votes
0answers
57 views

What applications does the theory of fibered surfaces have

Let $C$ be a smooth projective connected curve over $\mathbf{C}$. Let $X$ be a curve over the function field of $C$. Arakelov and Parshin proved the Mordell conjecture by considering a model for $X$ ...
0
votes
1answer
290 views

Where can I find a copy of Demazure and Gabriel's Introduction to algebraic geometry and algebraic groups

The question is pretty self explanatory. The book has been checked out of my university library, and I checked Amazon, and it says that the book is out of print. Also, I do not know French, so I am ...
11
votes
3answers
2k views

Best way to learn Algebraic Geometry?

I've been reading the book Commutative Algebra with a view towards Algebraic Geometry. I was wondering is the best way to learn algebraic geometry through commutative algebra? As the book I'm ...
2
votes
1answer
150 views

The reason for different terminologies

Different authors seem to have different conventions when they define the term affine variety (similarly projective variety). For the purposes of this question let us stick with the affine case, and ...