The study of geometric objects defined by polynomial equations, as well as their generalizations: algebraic curves, such as elliptic curves, and more generally algebraic varieties, schemes, etc. Problems under this tag typically involve techniques of abstract algebra or complex-analytic methods. ...

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17
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467 views

What problems, ideas or questions first got you interested in algebraic geometry?

Obviously, a lot of people are very interested in algebraic geometry. I suppose this means it is a fascinating area. However the few times I have tried to read introductory books and/or articles in ...
17
votes
1answer
416 views

Why is the rank of the Picard group of a K3 surface bounded above by 22?

I understand that, over $\mathbb{C}$, the rank of the Picard group of a K3 surface $X$ is bounded above by $20$ because we can use the exponential sheaf sequence: $0 \to 2\pi i \mathbb{Z} \to ...
17
votes
1answer
540 views

An exercise with Zariski topology

I read this exercise: Prove that the set $S = \{ (n, 2^n, 3^n ) \mid n \in \mathbb{N} \}$ is dense in $\mathbb{C}^3$ with Zariski topology. I have seriously thought about it, but I do not manage to ...
17
votes
2answers
943 views

History of Algebraic Geometry: Motivation behind definition of schemes

I am trying to read an article by Jean Dieudonne which talks about development of Algebraic Geometry. The article was being published in the journal "Advances in Mathematics" Volume 3, Issue 3, Pages ...
17
votes
3answers
401 views

What is the largest circle that fits in $\sin(x)?$

Imagine dropping a circle into the trough of $\sin(x)$. Would it reach the bottom or get wedged between two points on the curve? Depends on the size of the circle. So, what is the radius of the ...
16
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3answers
4k views

$\mathbb{A}^{2}$ not isomorphic to affine space minus the origin

Why is the affine space $\mathbb{A}^{2}$ not isomorphic to $\mathbb{A}^{2}$ minus the origin?
16
votes
2answers
964 views

What is algebraic geometry?

I am a second year physics undergrad, loooking to explore some areas of pure mathematics. A word that often pops up on the internet is algebraic geometry. What is this algebraic geometry exactly? ...
16
votes
4answers
3k views

Meaning of closed points of a scheme

This is a question in Liu's book. Let $X$ be a quasi-compact scheme. Show that $X$ contains a closed point. Well I'm unable to do this question, so any help would be appreciated. This question also ...
16
votes
3answers
653 views

Why was Sheaf cohomology invented?

Sheaf cohomology was first introduced into algebraic geometry by Serre. He used Čech cohomology to define sheaf cohomology. Grothendeick then later gave a more abstract definition of the right derived ...
16
votes
3answers
2k views

Hensel's Lemma and Implicit Function Theorem

In the literature and on the web happened to me several times to read confused or simply cryptic assertions regarding the fact that Hensel's Lemma is the algebraic version of Implicit Function ...
16
votes
7answers
1k 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 ...
16
votes
1answer
729 views

Homological methods in algebraic geometry

This question will probably seem quite silly to those well-versed in algebraic geometry (about which I admittedly hardly know anything); in the preface of Atiyah-Macdonald's book on commutative ...
16
votes
1answer
483 views

Is every rigid field perfect?

A field is rigid iff its automorphism group is trivial. A field $F$ is perfect iff all irreducibles in $F[x]$ are separable. Is every rigid field perfect?
16
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1answer
931 views

How to properly use GAGA correspondence

currently studying algebraic surfaces over the complex numbers. Before i did some algebraic geometry (I,II,start of III of Hartshorne) and a course on Riemann surfaces. Now i understood that by GAGA, ...
16
votes
1answer
401 views

What is a concrete example of why one wants to have a *derived category* in algebraic geometry?

My question asks for a concrete (and hopefully easy) example, why one wants to derive things in algebraic geometry. I heard, that a resolution of an object by free ones behaves much better than the ...
16
votes
1answer
702 views

When does variété mean manifold?

Following advice from this post, I am in the process of translating Ehresmann's 1934 paper "Sur la Topologie de Certains Espaces Homogènes" from French to English. French-English dictionaries online ...
16
votes
1answer
1k views

What is reductive group intuitively?

I am studying Geometric invariant theory and wonder how I should understand linearly reductive algebraic group. We say that an affine algebraic group $G$ is linearly reductive if all finite ...
16
votes
1answer
377 views

intuition on the projection formula

For a morphism $f:X\rightarrow Y$, locally free sheaf $\mathcal{G}$ on $Y$, and a quasi-coherent sheaf $\mathcal{G}$ on $X$, we have the projection formula ...
16
votes
1answer
457 views

Is there any holomorphic version of the tubular neighborhood theorem?

This question arised when I was studying Beauville's book 'Complex Algebraic Surfaces'. Castelnuovo's theorem says that a smooth rational curve $E$ on an algebraic surface $S$ is an exceptional ...
16
votes
1answer
357 views

Motivation for the study of amoebas.

What was the primary motivation for the study of the amoebas?
16
votes
1answer
334 views

What is the connection between Weil's character bound and Riemann Hypothesis over finite fields

Weil's character bound states that: Let $\mathbb{F}_{q}$ be a finite field of size $q$. Let $\chi$ be a multiplicative character of order $m$. Let $f(x)$ be a polynomial of degree $d$ such that $f(x) ...
16
votes
1answer
295 views

Formal Schemes Mittag-Leffler

Here is a question that is similar to my last one. I've been trying to learn about Grothendieck's Existence Theorem, but it seems that there aren't very many places that talk about formal schemes and ...
16
votes
1answer
196 views

A “generalized field” with $q$ elements, when $q$ is any number?

It is well-known that if a finite field has $q \in \mathbb{N}$ elements, then $q$ is prime power and $q > 1$. However, various modification of the concept of a "field" have been made in order to ...
15
votes
6answers
2k 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 ...
15
votes
2answers
890 views

Is tautological bundle $\mathcal{O}(1)$ or $\mathcal{O}(-1)$?

I always confused by whether tautological bundle is $\mathcal{O}(1)$ or $\mathcal{O}(-1)$, and definitions from different sources tangled in my brain. However, I thought this might not be simply a ...
15
votes
3answers
587 views

Connections between K-Theory and PDEs?

I've recently spent some time learning (the very basics of) K-theory for $C^*$-algebras and topological K-theory. Actually, my main fields of interest are PDEs and related topics, in particular ...
15
votes
3answers
3k 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 ...
15
votes
1answer
2k views

Why does the definition of an open subscheme / open immersion of schemes allow for an “extra” isomorphism?

After taking an algebraic geometry course last year, I've been reviewing the material this year, and I remembered something that struck me as odd, but which I'd neglected to ask about at the time: ...
15
votes
2answers
842 views

Intersection of Algebraic Topology/Geometry and Model Theory/Set Theory

Is there any intersection between the ideas of Algebraic Topology/Geometry (I know that there is most certainly a non-trivial intersection between Algebraic Geometry, Algebraic Topology, Arithmetic ...
15
votes
1answer
997 views

On limits, schemes and Spec functor

I have several related questions: Do there exist colimits in the category of schemes? If not, do there exist just direct limits? Do there exist limits? If not, do there exist just inverse limits? ...
15
votes
1answer
1k views

Proving that the genus of a nonsingular plane curve is $\frac{(d-1)(d-2)}{2}$

I'm studying from Joseph Silverman's book The Arithmetic Of Elliptic Curves and I'm trying to do as many exercises as I can. Right now I'm trying to do Exercise 2.7 from chapter II which reads as ...
15
votes
2answers
1k views

Theories of $p$-adic integration

What is the compelling need for introducing a theory of $p$-adic integration? Do the existing theories of $p$-adic integration use some kind of analogues of Lebesgue measures? That is, do we put a ...
15
votes
6answers
2k views

How do I teach university level mathematics to myself? [closed]

So here I go, I have enrolled myself in maths major this year but due to less marks in SSC I couldn't secure admission in a good university so I have to take admission wherever I could get with my ...
15
votes
2answers
766 views

Motivating (iso)morphism of varieties

I am reading course notes on algebraic geometry, where a morphism of varieties is defined as follows ($k$ is an algebraically closed field): Let $X$ be a quasi-affine or quasi-projective ...
15
votes
1answer
627 views

Hilbert's Original Proof of the Nullstellensatz

Does anyone have a link to Hilbert's Original Proof of the Nullstellensatz, or know a book where it's printed? I'd be interested to see what it was like. I only really know the Noether normalisation ...
15
votes
4answers
574 views

Isomorphism between quotient rings of $K[X,Y]$

Let $K$ be a field of characteristic $0$ and $m,n\in\mathbb Z$, $m,n\ge 1$. Prove that $$K[X,Y]/(X^2-Y^m)\simeq K[X,Y]/(X^2-Y^n)$$ if and only if $m=n$. (Related to Isomorphism between quotient rings ...
15
votes
1answer
695 views

what is a “dévissage” argument?

In Serre's "Local Fields" (Chapter 2, section 2, proposition 3) he proves something about field extensions which he breaks into parts: first he dealt with the separable case, and then with the ...
15
votes
1answer
816 views

vector bundles on affine schemes

Serre's theorem (one of them) states that for a quasi-coherent sheaf $\mathscr F$ on an affine noetherian scheme $H^i(X,\mathscr{F})$ vanish for $i >0$. I used to think that this would imply that ...
15
votes
1answer
339 views

Definition(s) of Stack

I've started to learn about stacks, and a question arose in my attempts of looking at the very definition of a stack by several points of view. First, I recall some background and fix the notation ...
15
votes
1answer
436 views

principal G-bundles in zariski vs etale topology

Let $G$ be an (affine) algebraic group over say $\mathbb{C}$. A principal $G$-bundle is a scheme $P$ with a $G$ action and a $G$-invariant morphism of schemes $\pi:P \to X$ that is etale locally on ...
15
votes
1answer
264 views

What is the importance of modules in algebraic geometry?

I have been trying to teach myself the basics of algebraic geometry. I understand the basic premise, how we define geometry spaces (algebraic sets and schemes) in terms of commutative rings. And I ...
15
votes
1answer
320 views

Global Optimization and Real Algebraic Geometry

Wikipedia suggests that: "Methods based on real algebraic geometry" are some of the "most successful general strategies" for solving global optimization problems. Could someone suggest an reference ...
15
votes
1answer
285 views

Hypercube and Hyperspheres

Let $n,k\in\mathbb{N}$. In this problem, the geometry of $\mathbb{R}^n$ is the usual Euclidean geometry. The lattice hypercube $ Q(n,k)$ is defined to be the set $ \{1,2,...,k\}^n ...
15
votes
3answers
629 views

*writing* proofs involving commutative diagrams

This question is a little fuzzy so might be closed, but I'll give it a shot. I'm sorry this question has quite a long introduction, I don't see how to formulate it more concisely. In modern algebraic ...
15
votes
1answer
286 views

A space of ideals

Definition: Let $R$ be a commutative ring with 1. Endow the power set $2^R$ with the product topology. The ideal space $\mathcal{I}(R)$ is defined to be subset of $2^R$ consisting of ideals, ...
15
votes
1answer
313 views

Understanding proof by algebraic geometry, Fermat's last theorem for polynomials when $n = 3$.

This is a followup to my question here. See here. The question is as follows. How do we see that there do not exist nonconstant, relatively prime, polynomials $a(t)$, $b(t)$, and $c(t) \in ...
15
votes
1answer
197 views

Is there a geometric meaning of a prime power not being primary?

I guess that the standard example of a prime power that is not a primary ideal is $$\mathfrak p^2 :=(x,z)^2\subset k[x,y,z]/(xy-z^2):=A.$$ Because $\mathfrak p^2 = (x^2,xz,xy)$, we see that $x\not ...
15
votes
1answer
294 views

A Genetic Introduction to Algebraic Geometry

I am a big fan of "genetic introductions" in mathematics, i.e. where the ideas are introduced in the order they were developed along with why they were introduced, as opposed to the ...
15
votes
0answers
205 views

Complex manifold with subvarieties but no submanifolds

Note, I have now asked this question on MathOverflow. There are examples of compact complex manifolds with no positive-dimensional compact complex submanifolds. For example, generic tori of ...
14
votes
4answers
1k views

Picard group of product of spaces

Suppose $X,Y$ are varieties over an algebraically closed field $k$. Can we compute $\operatorname{Pic}(X \times_k Y) $ in terms of $\operatorname{Pic}(X),\operatorname{Pic}(Y)$? It seems that ...