# Tagged Questions

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### First Description of how to Remove Radicals from Equations

Who first described the technique of removing radicals as indicated in the answers to questions Tools for Removing Radicals from Equations and Rewrite sum of radicals equation as polynomial equation ? ...

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651 views

### Salvaging Leibnizian formalism?

Can one justify Leibniz's formalism in a suitable algebraic or topological context?
We have published some papers recently where we argue that Leibniz's formalism for the calculus wasn't ...

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683 views

### History of the connection between Riemann surfaces and complex algebraic curves

As noted in the question "Links between Riemann surfaces and algebraic geometry", there are strong connections between Riemann surfaces and algebraic geometry - for example, compact Riemann surfaces ...

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**0**answers

464 views

### History of the Proj construction in algebraic geometry

Projective geometry was introduced by fifteenth century Renaissance painters (like Alberti, da Vinci and Dürer) in the guise of perspective theory, although one could argue that Pappus was already ...

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**2**answers

287 views

### Origin of the name “Torelli group”

The genus $g$ Torelli group $I_g$ is the kernel of the action of the mapping class group of a genus $g$ surface on the first homology group of the surface.
The first paper I am aware of that uses the ...

**12**

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1k views

### History and motivation for Tannaka, Krein, Grothendieck, Deligne et al. works on Tannaka-Krein theory?

I am trying to wrap my mind around Tannaka-Krein duality and it seems quite mysterious for me, as well, as its history. So let me ask:
Question: What was the motivation and historical context for ...

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907 views

### Deligne Weil II

Deligne's Weil I has been published under the title "La conjecture de Weil: I" in 1974, and Weil II in 1980. So did Deligne know in 1974 that there would be a Weil II, and can one explain the period ...

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**1**answer

363 views

### What was the original/historical motivation for introducing Grothendieck (pre-)topologies

The title essentially explains it, but I'll give some background:
I'm giving a talk to some fellow grad students about the relative Picard functor which requires introducing Grothendieck ...

**22**

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**3**answers

938 views

### What exactly does this diagram of Omar Khayyam represent?

Evidently Omar Khayyam (1048-1131) was quite the mathematician. He did groundbreaking work on finding geometric solutions to the cubic equation, which is all the more notable since he did not have a ...

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481 views

### To what extent can fields be classified?

The study of algebraic geometry usually begins with the choice of a base field $k$. In practice, this is usually one of the prime fields $\mathbb{Q}$ or $\mathbb{F}_p$, or topological completions and ...

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377 views

### Biholomophic non-Algebraically Isomorphic Varieties

Recently, when writing a review for MathSciNet, the following question arose:
Is it true that two smooth complex varieties that are biholomorphic are algebraically isomorphic? The converse is true ...

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**1**answer

357 views

### Generalization of the Lefschetz fixed point theorem

I have encountered a certain generalization of the Lefschetz fixed point theorem as folklore, and I am hoping that someone out there knows its provenance or can otherwise refer me to a source where it ...

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815 views

### who invented projective space $\mathbb{P}^n$?

Who invented projective space $\mathbb{P}^n$ as an extension of the usual affine space $\mathbb{A}^n$?
Who was the first person to consider projective closure of
plane affine algebraic curves (curves ...

**15**

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**0**answers

417 views

### Authorship of Grothendieck universes

Universes seem to first appear in Grothendieck's work in SGA 1, which is credited to Grothendieck, and a lengthy discussion is in the chapter on Prefaisceaux (presheaves) in SGA 4. That chapter is ...

**8**

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**1**answer

2k views

### What is “Teichmüller Theory” and its history ?

What is "Teichmüller Theory" ? What part has been worked out / forseen by O. Teichmüller himself and what is further development ? Is there some current work which might be considered as ...

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617 views

### Why is Gauss credited with his connection?

Let $\pi : X \to B$ be a family of compact Kähler manifolds over a smooth base $B$. We then have a local system $\mathcal R^k \pi_* \mathbb Z$ (for your favorite $k$) of abelian groups over $B$, whose ...

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434 views

### Origin of the theorem on the existence of the smallest field of definition of an affine variety

Weil proved the following theorem in his book Foundations of Algebraic Geometry, p.19.
The proof is somewhat involved.
I wonder if the theorem is his original.
Theorem
Let $K[X_1,\dots, X_n]$ be the ...

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**2**answers

1k views

### Heuristics for the Hodge Conjecture

W. V. D. Hodge is famous for his Hodge conjecture, one of the Millennium prize problems. Hodge might have had some rough heuristics or ideas that led him to the formulation of the conjecture.
I am ...

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1k views

### Why the letter “p” for genus?

Does anybody know why the genus (arithmetic or geometric) of a curve was historically denoted by $p$ ($p_a$ and $p_g$)? What does the letter "$p$" stand for?
Any references would be greatly ...

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**1**answer

784 views

### Who first cared about singular points?

If you look at the cross $C\subset \mathbb A^2_k$ given by $xy=0$ in the affine plane over the field $k$, you see or compute that it is exceptional at $O=(0,0)$ for many (obviously not independent) ...

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205 views

### Key variety technique, a history question

I have a history question about the technique called "key variety technique" used in algebraic geometry. (see eg http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.9.6880). One can find many ...

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991 views

### Did Grothendieck introduce vertical arrows that denote morphisms?

It is usual in algebraic geometry to represent morphisms by vertical arrows pointing downwards, like that :
$$\begin{matrix} X \\\\ \downarrow \\\\ S \end{matrix}$$
I suppose this stemmed from ...

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**1**answer

1k views

### State of resolution in positive characteristic?

Heisuke Hironaka's coming talk makes me wonder how the state of the work on that theme is. So far, I noticed (but didn't read) these programs: 1, 2. It would be great if someone who listenes ...

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3k views

### Origin of terms “flag”, “flag manifold”, “flag variety”?

These terms have become common in Lie theory and related algebraic geometry and combinatorics, as seen in many questions posted on MO, but it's unclear to me where they first came into use. Probably ...

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703 views

### Historical Articles about zeta functions of curves

Are there any historical articles about the origins of zeta functions of curves over global fields (undoubtedly starting with $\mathbb{Q}$)? In particular who (and when did this happen) first have ...

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**1**answer

887 views

### Why and how did preschemes become schemes?

Originally (e.g., in the first edition of EGA and in Mumford's Red Book), what are now called "schemes" were referred to as "preschemes." The word "scheme" was reserved for what are now called ...

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**1**answer

1k views

### How were moduli spaces defined before functors?

People today in algebraic geometry will typically define a moduli space to be the space which represents the functor of families of whatever object they are interested in studying.
However, I am ...

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3k views

### where can you find Grothendieck's “Recoltes et Semailles”?

Where can you find Grothendieck's "Recoltes et Semailles"?
Is it available anywhere?

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4k views

### Has the mathematical content of Grothendieck's “Récoltes et Semailles” been used?

This question is partly motivated by this one.
Motivation
Grothendieck's "Récoltes et Semailles" has been cited on various occasions on this forum. See for instance the answers to this question or ...

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**0**answers

502 views

### Missing exposes in SGA 5, and the composition of the SGA's

Over the past couple of years I had to look in SGA for various results, and I can't but marvel at how poorly constructed it is. In SGA1 expose VII "n'existe pas", SGA 1 references higher SGA's, and so ...

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649 views

### Why are people interested in defining GW invariant in algebraic geometry category

Originally it is in symplectic geometry. Is it just curosity or any other special reason? Thank you for clarifying.

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**1**answer

920 views

### Why “syntomic” if “flat, locally of finite presentation, and local complete intersection” is already available?

Dear everyone,
(i) Who is the father of the adjective “syntomic” in algebraic geometry?
(ii) And why did he choose to introduce a new term for what we already know from EGA IV.19.3.6 and SGA ...

**16**

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**3**answers

1k views

### Origin of the term “localization” for the localization of a ring

I'm curious if the term localization in ring theory comes from algebraic geometry or not. The connection between localization and "looking locally about a point" seems like it should be the source ...

**35**

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**4**answers

3k views

### Did Gelfand's theory of commutative Banach algebras influence algebraic geometers?

Guillemin and Sternberg wrote the following in 1987 in a short article called "Some remarks on I.M. Gelfand's works" accompanying Gelfand's Collected Papers, Volume I:
The theory of commutative ...

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**6**answers

7k views

### what mistakes did the Italian algebraic geometers actually make?

It's "well-known" that the 19th century Italian school of algebraic geometry made great progress but also started to flounder due to lack of rigour, possibly in part due to the fact that foundations ...

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574 views

### A historical question: Hurwitz, Luroth, Clebsch, and the connectedness of M_g

The connectedness of the moduli space M_g of complex algebraic curves of genus g can be proven by showing that it is dominated by a Hurwitz space of simply branched d-fold covers of the line, which in ...

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**2**answers

2k views

### Italian school of algebraic geometry and rigorous proofs

Many of the amazing results by Italian geometers of the second half of the 19th and the first half of the 20th century were initially given heuristic explanations rather than rigorous proofs by their ...

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2k views

### Wild Ramification

The question is, loosely put, what is known about wild ramification?
Is there a semi-well-established theory of wild ramification that can be furthered in various specific situations? Or maybe there ...

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895 views

### Classical Enumerative Geometry References

I want to start out by making this clear: I'm NOT looking for the modern proofs and rigorous statements of things.
What I am looking for are references for classical enumerative geometry, back before ...

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2k views

### Do there exist modern expositions of Klein's Icosahedron?

Reading Serre's letter to Gray
, I wonder if now modern expositions of the themes in Klein's book
exist. Do you know any?