The study of geometric objects defined by polynomial equations. Algebraic curves, such as elliptic curves, and more generally algebraic varieties, schemes, etc.
157
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6answers
4k views
“The Egg:” Bizarre behavior of the roots of a family of polynomials.
In this MO post, I ran into the following family of polynomials: $$f_n(x)=\sum_{m=0}^{n}\prod_{k=0}^{m-1}\frac{x^n-x^k}{x^m-x^k}.$$
In the context of the post, $x$ was a prime number, and $f_n(x)$ ...
49
votes
1answer
2k views
Trigonometric sums related to the Verlinde formula
Original question (see also the revised, possibly simpler, version below): Let $g > 1, r > 1$ be integers. Playing around with the Verlinde formula (see below), I came across the expression
...
40
votes
2answers
3k 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 ...
32
votes
1answer
527 views
Geometric intuition behind The Mordell Conjecture
The Mordell Conjecture/Faltings Theorem says roughly that if $K$ is an algebraic number field and $X$ is an algebraic curve defined over $K$ of genus $g >1$ then the set of $K$-rational points ...
30
votes
7answers
571 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 ...
28
votes
1answer
684 views
Is there an atlas of Algebraic Groups and corresponding Coordinate rings?
I was wondering if there was a resource that listed known algebraic groups and their corresponding coordinate rings.
Edit: The previous wording was terrible.
Given an algebraic group $G$, with Borel ...
25
votes
1answer
1k views
Sheaf cohomology: what is it and where can I learn it?
As I understand it, sheaf cohomology is now an indispensable tool in algebraic geometry, but was originally developed to solve problems in algebraic topology. I have two questions about the matter.
...
23
votes
3answers
658 views
Geometry or topology behind the “impossible staircase”
This question on the topology of Escher games reminded me of a question I've had in my head for a little while now.
Is there anything interesting geometric or topological that can be said about the ...
23
votes
5answers
1k views
Help understanding Algebraic Geometry
I while ago I started reading Hartshorne's Algebraic Geometry and it almost immediately felt like I hit a brick wall. I have some experience with category theory and abstract algebra but not with ...
22
votes
5answers
1k views
Why should I care about adjoint functors
I am comfortable with the definition of adjoint functors. I have done a few exercises proving that certain pairs of functors are adjoint (tensor and hom, sheafification and forgetful, direct image and ...
22
votes
3answers
537 views
Why the emphasis on Projective Space in Algebraic Geometry?
I have no doubt this is a basic question. However, I am working through Miranda's book on Riemann surfaces and algebraic curves, and it has yet to be addressed.
Why does Miranda (and from what little ...
22
votes
1answer
497 views
Why is the Hessian of an irreducible polynomial not zero?
Let $k$ be an algebraically closed field, $\operatorname{char}k=0$, $F$ be an irreducible homogeneous polynomial of degree$>1$ in $k[X,Y,Z]$, and ...
21
votes
10answers
3k views
Best Algebraic Geometry text book? (other than Hartshorne)
Lifted from Mathoverflow:
I think (almost) everyone agrees that Hartshorne's Algebraic Geometry is still the best.
Then what might be the 2nd best? It can be a book, preprint, online lecture note, ...
21
votes
5answers
1k views
Famous papers in algebraic geometry
I'm reading the Mathoverflow thread "Do you read the masters?", and it seems the answer is a partial "yes".
Some "masters" are mentioned, for example Riemann and Zariski. In particular, a paper by ...
21
votes
3answers
468 views
Why is it “easier” to work with function fields than with algebraic number fields?
I just bought a copy of Jürgen Neukirch's book Algebraic Number Theory. While browsing through it I found a section titled § 14. Function Fields in chapter I. In it the author describes ...
21
votes
1answer
561 views
Deligne, elliptic curves and modular forms
I'm trying to understand an argument of Deligne (in Courbes elliptiques: Formulaire d'après J. Tate), but I'm not familiar enough with algebraic geometry, so I'm getting quite confused. So even in my ...
21
votes
1answer
407 views
Quasi-coherent sheaves, schemes, and the Gabriel-Rosenberg theorem
In the context of commutative rings, a ring is completely determined by its category of modules. That is, two commutative rings $R$ and $S$ are isomorphic if and only if the category of $R$-modules is ...
21
votes
1answer
442 views
functoriality of derivations
I seem to have problems understanding algebraically why given a map of manifolds $f: M \to N$ we get a bundle map $TM \to f^*TN$.
Now, fiberwise it's all good. But I do not understand how to define ...
20
votes
2answers
914 views
How many cubic curves are there?
It is well-known that there is only one "kind" of line, and that there are three "kinds" of quadratic curves (the nature of which depends on the sign of a so-called "discriminant").
It is noteworthy ...
20
votes
2answers
606 views
How do different definitions of “degree” coincide?
I've recently read about a number of different notions of "degree." Reading over Javier Álvarez' excellent answer for the thousandth time finally prompted me to ask this question:
How exactly do ...
19
votes
1answer
346 views
Connectedness of the spectrum of a tensor product.
Let $A$, $B$ be finite, free $\mathbb{Z}$ algebras such that $\operatorname{Spec}(A)$ and $\operatorname{Spec}(B)$ are both connected. Is $\operatorname{Spec}(A\otimes_{\mathbb{Z}} B)$ connected?
19
votes
1answer
380 views
An interesting topological space with $4$ elements
There is an interesting topological space $X$ with just four elements $\eta,\eta',x,x'$ whose nontrivial open subsets are $\{\eta\},\{\eta'\},\{\eta,\eta'\}, \{\eta,x,\eta'\}, \{\eta,x',\eta'\}$. This ...
18
votes
2answers
424 views
Who first called the Grothendieck's schéma scheme?
Grothendieck called "schemes" schémas in French.
I find it strange we call them schemes.
In fact, Grothendieck called them (pre-) schemas(this is an English word) in his talk(in English) at Proceeding ...
18
votes
0answers
486 views
Grothendieck 's question - any update?
I was reading Barry Mazur's biography and come across this part:
Grothendieck was exceptionally patient with me, for when we first met I knew next to nothing about algebra. In one of his first ...
18
votes
0answers
261 views
What is the Picard group of $z^3=y(y^2-x^2)(x-1)$?
I'm actually doing much more with this affine surface than just looking for the Picard group. I have already proved many things about this surface, and have many more things to look at it, but the ...
17
votes
2answers
514 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? ...
17
votes
4answers
1k views
Motivating Example for Algebraic Geometry/Scheme Theory
I am in the process of trying to learn algebraic geometry via schemes and am wondering if there are simple motivating examples of why you would want to consider these structures.
I think my biggest ...
17
votes
4answers
943 views
What are the required backgrounds of Robin Hartshorne's Algebraic Geometry book?
It seems that Robin Hartshorne's Algebraic Geometry is the place where a whole generation of fresh minds have successfully learned about the modern AG. But is it possible for someone who is out of the ...
17
votes
2answers
769 views
Original works of great mathematician Évariste Galois
Through this question I wanted to know the original works of Galois. When I was reading Galois theory ( since from last month ) , I have been seeing one common line in every book, whose essence ...
17
votes
5answers
2k views
Using Gröbner bases for solving polynomial equations
In my attempts to understand just how computer algebra systems "do things", I tried to dig around a bit on Gröbner bases, which are described almost everywhere as "a generalization of the Euclidean ...
17
votes
0answers
293 views
cones in the derived category
If I have two exact triangles $X \to Y \to Z \to X[1]$ and $X' \to Y' \to Z' \to X'[1]$ in a triangulated category, and I have morphisms $X \to X'$, $Y \to Y'$ which 'commute' ...
16
votes
5answers
870 views
Why Zariski topology?
Why in algebraic geometry we usually consider the Zariski topology on $\mathbb A^n_k$? Ultimately it seems a not very interesting topology, infact the open sets are very large and it doesn't satisfy ...
16
votes
4answers
1k views
intuitive explantions for the concepts of divisor and genus
when trying to explain AG-codes to computer scientists, the major points of contention i am faced with are the concepts of divisors, Riemann-Roch space and the genus of a function field. are there any ...
16
votes
2answers
379 views
The prime spectrum of a Dedekind Domain
Let $A$ be a Dedekind Domain, let $X = \operatorname{Spec}(A)$. Are all open sets in $X$ basic open sets? Thinking about the Zariski topology (in the classical sense) of a non-singular affine curve, ...
16
votes
2answers
254 views
When is the sheaf corresponding to a vector bundle on a smooth manifold coherent?
In algebraic and analytic geometry, vector bundles are usually interpreted as locally free sheaves of modules (over the structure sheaves). They are in particular examples of quasi-coherent sheaves. ...
16
votes
1answer
212 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
0answers
166 views
Tate conjecture for Fermat varieties
I've been looking at Tate's Algebraic Cycles and Poles of Zeta Functions (hard to find online... Google books outline here) and have a question about his work on (conjecturing!) the Tate conjecture ...
15
votes
4answers
663 views
Why isn't $\mathbb{C}[x,y,z]/(xz-y)$ a flat $\mathbb{C}[x,y]$-module
Why isn't $M = \mathbb{C}[x,y,z]/(xz-y)$ a flat $R = \mathbb{C}[x,y]$-module?
The reason given on the book is "the surface defined by $y-xz$ doesn't lie flat on the $(x,y)$-plane". But I don't ...
15
votes
3answers
1k views
What use is the Yoneda lemma?
Although I know very little category theory, I really do find it a pretty branch of mathematics and consider it quite useful, especially when it comes to laying down definitions and unifying diverse ...
15
votes
2answers
509 views
Did Zariski really define the Zariski topology on the prime spectrum of a ring?
The question is not: “Did Zariski really define the Zariski topology?”
It is: “Did Zariski really define the Zariski topology on the prime spectrum of a ring?”
Here is the motivation. --- On page ...
15
votes
4answers
389 views
Material in a first course in algebraic geometry?
First I would like to say that my question is not about what books to use in algebraic geometry; for this there are many threads that discuss this on Math.SE and on MO. My question is about what ...
15
votes
1answer
257 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, ...
14
votes
8answers
3k views
(undergraduate) Algebraic Geometry Textbook Recomendations
What are the best algebraic geometry textbooks for undergraduate students?
14
votes
1answer
607 views
Why is the hard Lefschetz theorem “hard”?
Let $X$ be a compact Kähler manifold of complex dimension $\dim_{\mathbb C} = n$. Let $[\omega]$ be the cohomology class of a Kähler metric on $X$. Then powers of the class $[\omega]$ defines a linear ...
14
votes
3answers
581 views
What is a local parameter in algebraic geometry?
Shafarevich offers the following theorem-definition:
"At any nonsingular point $P$ of an irreducible algebraic curve, there exists a regular function $t$ that vanishes at $P$ and such that every ...
14
votes
1answer
309 views
Can an integral scheme have closed points of both positive and zero characteristic?
Background
Recall that an integral scheme $X$ is a scheme which is both irreducible and reduced; equivalently, its ring of functions is an integral domain on every open subset.
Given any point $p$, ...
14
votes
1answer
360 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 ...
14
votes
1answer
397 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 ...
14
votes
1answer
219 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 ...
14
votes
3answers
320 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 ...
