Questions tagged [algebraic-geometry]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
2 votes
1 answer
186 views

Grothendieck and Fields medal 1962

We can read as a mathunion excerpt that Grothendieck won the Fields medal in 1966 Built on work of Weil and Zariski and effected fundamental advances in algebraic geometry. He introduced the idea of ...
user avatar
2 votes
0 answers
76 views

The Picard Group: Origin and History

I've come to the notion of the Picard Group. I was recently linked to this paper paper, which contains the line: The problem of computing the Picard groups of surfaces $S \subset \mathbb{P}_{\mathbb{...
user avatar
2 votes
0 answers
111 views

Did the ancients construct higher genus curves?

I know that the ancients had several ways to construct geometric shapes. I know two of them: the biggest one, ruler and compass, with which you can build some polygons, bisect an angle, etc, but not ...
user avatar
3 votes
1 answer
109 views

Why is the term "isotropic" used to describe a quadratic form and a vector?

A quadratic form $q$ on a vector space $V$ is isotropic if $q(v) = 0$ admits a nonzero solution. What was this terminology originally intended to evoke? There is some prior discussion on Math SE, ...
user avatar
2 votes
0 answers
82 views

Continuation on Galois’ lost memoir(s): was Poisson right in his review of Galois’ memoir?

A recent question asked whether Galois’ “lost” memoir has been since recovered (similar to Abel’s lost memoir). The memoir referenced in the question is the one Galois submitted to the French “...
user avatar
  • 151
0 votes
1 answer
66 views

Who first started parameterizing hyperbolas with hyperbolic functions?

Anyone know the history of using hyperbolic functions to parameterize the parabola? Wolfram hasn't really helped. Anyone know where to look?
user avatar
  • 109
3 votes
1 answer
195 views

Origins of Zariski topology

Why did Zariski feel the need to define his famous topology? Was this notion used in one form or another prior to him in algebraic geometry?
user avatar
  • 151
9 votes
0 answers
223 views

First use of term "Hilbert's Nullstellensatz"

This year (2021) marks the 100th anniversary of Emmy Noether's 1921 paper in which she introduced Noetherian rings and proved the primary ideal decomposition for them. The original version of her ...
user avatar
  • 4,064
3 votes
1 answer
115 views

Complete list of publications of Rebecca Barlow

Rebecca Barlow is the discoverer of an interesting surface in algebraic geometry. Is anybody aware of a full list of her contributions? Has she continued working in mathematics in the 21st century?
user avatar
  • 31
9 votes
1 answer
476 views

What is the etymology of the mathematical terms "sheaf, stalk, germ"?

The peculiar agricultural terminology commonly used in algebraic geometry and category theory, "sheaf", "stalk", "germ", is quite well-known. A sheaf is pictured as something like a bundle of stalks, ...
user avatar
6 votes
1 answer
548 views

What was the motive for inventing Gröbner bases?

How did professor Buchberger discover Gröbner (Groebner) bases for polynomial ideals? What was the problem(s) that lead to such a discovery?
user avatar
5 votes
2 answers
339 views

Who first described the fundamental group as the group of deck transformations?

Grothendieck developed the theory of the fundamental group of a scheme in SGA 1. In order to do so he used the fact that the fundamental group of a topological space is isomorphic to the group of deck ...
user avatar
8 votes
2 answers
307 views

(Co)Homology: From topology to the rest of mathematics?

I can appreciate how (co)homology arose in the context of topology/geometry. Trying to get a handle on the handles of spaces leads one to this idea. It's not obvious, but I can see how this would lead ...
user avatar
2 votes
1 answer
127 views

Material models of Riemann surfaces

It is known that during the last quarter of the 19th century there was a flourishing of the production of material models (from plaster, strings, card-board etc) of curves and surfaces in Germany (but ...
user avatar
  • 283
2 votes
3 answers
222 views

Who first had the idea to study surfaces via rings of functions, as in algebraic geometry?

This idea provides the foundations of algebraic geometry now; and they have certainly gone down the rabbit hole with it. As a student studying this subject, I have always found it such a great leap to ...
user avatar
20 votes
4 answers
3k views

Why are étale morphisms called "étale"?

Alexander Grothendieck developed the theory of "locally trivial coverings spaces for rings/schemes" in SGAI as an analog to the theory of covering spaces in algebraic topology. He called such ...
user avatar
3 votes
2 answers
201 views

When did mathematician start to draw figures from equation?

I know that when solving geometric problem, Descartes used variables $x,y$ and derived equation such as $y^2=cy-\frac{cxy}{b}+ay-ac$. Conversely, in algebraic geometry, an arbitrary polynomial $F(X_1,....
user avatar
11 votes
2 answers
1k views

Where was the word "pencil" first used in (projective) geometry and what is the reason behind this curious name?

The title is pretty self-explanatory: A pencil in projective (or algebraic) geometry is the family of all lines through a point. The above-linked website tells me that Cremona, on page x of Elements ...
user avatar
  • 3,681
15 votes
3 answers
1k views

How did Grothendieck encounter and adopt the categorical language?

Apparently the first mathematical publication of Grothendieck where he uses the terms “functor” and “category” in the technical sense is the Kansas report. From where Grothendieck had knowledge of ...
user avatar
  • 542
5 votes
1 answer
169 views

Origins and history of branched covering

During my research on branched coverings of the projective plane, I am interested to know the origins and history of branched coverings of the projective plane and the projective line, together with ...
user avatar
  • 51
6 votes
1 answer
311 views

Grothendieck and elementary topos

I would like to know some references (if any) for the claim that Grothendieck didn't like the idea of elementary topos.
user avatar
  • 542
7 votes
1 answer
975 views

Visualizing algebra before Descartes

The Cartesian coordinate system is what I have been told provided the first link between algebra and geometry. However, I also have learned that, for instance, Omar Khayyam solved cubic equations as ...
user avatar
  • 367
16 votes
3 answers
3k views

Grothendieck's approach to solving problems

Alexander Grothendieck is known to have revolutionized several areas of mathematics. His insights were very deep, original and revolutionary. Time after time he showed that he could see in ways that ...
user avatar
  • 169
16 votes
1 answer
796 views

Who first introduced the notation $\mathcal{O}$ in algebraic geometry or algebraic number theory

This is my first question for HSM. If it is consider too specialized for HSM, perhaps it can be migrated to MathOverflow. In algebraic number theory, one frequently denotes the ring of algebraic ...
user avatar
5 votes
0 answers
198 views

History of Algebraic Geometry for a general readership?

I'm looking for a "pop maths" book on the subject. Something much more accessible than Dieudonné: "light reading" with emphasis on the history, personalities and general ideas and minimal ...
user avatar
9 votes
3 answers
2k views

What is the history behind the concept of "schemes" in algebraic geometry?

Let me start by saying that I've just started studying algebraic geometry. I have a habit of trying to find out something about the motivation behind new concepts that I'm studying, in this case that ...
user avatar
16 votes
1 answer
311 views

Did Dedekind show any evidence of pictorial geometric sense?

The most pictorial geometric thinking I find in Dedekind is the intuitive-geometric idea of a continuum which he criticizes as too vague before he gives his account of the continuum based on cuts. ...
user avatar
40 votes
1 answer
8k views

Why did algebraic geometry need Alexander Grothendieck?

Grothendieck is arguably the most brilliant mathematician of the 20th century, with his influence felt the most in algebraic geometry, which he transformed. Some time ago the story used to be told was ...
user avatar
  • 66.2k