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Shafarevich 『Basic Algebraic Geometry 1』序文註解

\[ %%% 黒板太字 %%% \newcommand{\A}{\mathbb{A}} %アフィン空間 \newcommand{\C}{\mathbb{C}} %複素数 \newcommand{\F}{\mathbb{F}} %有限体 \newcommand{\N}{\mathbb{N}} %自然数 \newcommand{\Q}{\mathbb{Q}} %有理数 \newcommand{\R}{\mathbb{R}} %実数 \newcommand{\Z}{\mathbb{Z}} %整数 %%% 2項演算 %%% \newcommand{\f}[2]{ \frac{#1}{#2} } \]

はじめに

本稿ではShafarevich 『Basic Algebraic Geometry 1』の初版に対する序文を読み解いていく.といっても数学的な内容を補うわけではない.行うのは主に英語の読解である.

なぜこういうことをするのかというと,私の英語の勉強のためである.だから英和辞書は使用せず英英辞典を使っている.あともうひとつ目的があって,それは数学書の序文だけでもしっかり読むことで,なんとなくその本を読破したかのような気持ちになることである.特に初めて学ぶ分野の場合,序文を読んだだけでもその分野の概略がなんとなく判るので有用だと思う.

使用させていただいたツールを明記しておく.

まず辞書としては Cambridge dictionary の英英辞典を使用した.

和訳を確認する必要があるときには DeepL を参照した.本稿で紹介した和訳は DeepL と相談しながら推敲している.

例文や使用頻度,共起表現を確認するためにコーパスとして SKELL を使用した.

いずれも無料で使用させていただけてありがたい.

第1段落

Algebraic geometry played a central role in 19th century math. The deepest results of Abel, Riemann, Weierstrass, and many of the most important works of Klein and Poincaré were part of this subject.

1文目

Algebraic geometry played a central role in 19th century math.

2文目

The deepest results of Abel, Riemann, Weierstrass, and many of the most important works of Klein and Poincaré were part of this subject.

和訳

Algebraic geometry played a central role in 19th century math.

代数幾何学は,19世紀の数学において中心的な役割をもっていた.

The deepest results of Abel, Riemann, Weierstrass, and many of the most important works of Klein and Poincaré were part of this subject.

Abel や Riemann や Weierstrass が得た最も深い結果,そして Klein や Poincaré の重要な仕事の大半が,代数幾何に属するものだった.

第2段落

The turn of the 20th century saw a sharp change in attitude to algebraic geometry. In the 1910s Klein writes as follows: “In my student days, under the influence of the Jacobi tradition, Abelian functions were considered as the unarguable pinnacle of math. Every one of us felt the natural ambition to make some independent progress in this field. And now? The younger generation scarcely knows what Abelian functions are.” (From the modern viewpoint, the theory of Abelian functions is an analytic aspect of the theory of Abelian varieties, that is, projective algebraic group varieties; compare the historical sketch.)

1文目

The turn of the 20th century saw a sharp change in attitude to algebraic geometry.

saw a change

the turn of the 20th century を主語とした無生物主語文になっている.同様に時期や時代を主語とする例文に,

  • January 1837 saw a change in the war.

  • The 1970s saw a change in attitude and population.

  • The Second World War saw a change in fortunes.

  • The 1930s also saw a change in the role of government.

などがある.したがって,与えられた文を attitude to algebraic geometry を主語として

Attitude to algebraic geometry changed sharply in the turn of the 20th century.

などと書き直しても主述関係は変わらないだろうことが判る.

attitude

attitudeというのはbehaviorやbeliefやideaの類語で,おそらく

  • a feeling or opinion about something or someone, or a way of behaving that is caused by this

という意味で使われている.

語尾の-itudeは「性質・状態・質」を指す語根であり,抽象的な名詞をつくる.たとえば,gratitude や magnitude や solitude なども同じ語根を持つ.

turn

turnというのは,名詞に限れば

  • Time to do something: an opportunity or a duty to do something.

  • Go round

  • Change in direction

  • Development

  • Performance: a stage act or performance

  • Illness: a strange feeling, or a nervous shock

という意味を持つ.the turn of the century という慣用句があり,これには 「the time when a particular century ends and another begins」という意味がある.

sharp

sharpという単語は,ここでは形容詞なので

  • able to cut

  • sudden: happening suddenly, quickly, and strongly

  • taste: sour in taste

  • clear: easy to see or understand

  • clever

  • fashionable

という意味を持つ.ここではsharp changeと言っているので,suddenが近いだろう.

2文目

In the 1910s Klein writes as follows:

3文目

“In my student days, under the influence of the Jacobi tradition, Abelian functions were considered as the unarguable pinnacle of math.

unarguable

unarguableというのは,

  • It is unarguable that quality is much better than quantity.

というように使う言葉であり,「 not able to be argued 」という意味である.関連語に clear, indeed などがある.

arguable の頭に否定を意味する un- がついた言葉である.Σ が arguable であるというのは,there could be some disagreement about Σ ということを意味する.

pinnacle

pinnacleという単語は,ここでは名詞として使われているので

  • success: the most successful or admired part of a system or achievement

  • top: a small pointed tower on top of a building, or the top part of a mountain

という意味である.

なお語尾についている -cle は particle, circle, chronicle, miracle などについている -cle と元が同じである.

4文目

Every one of us felt the natural ambition to make some independent progress in this field.

independent

independent という単語は,ここでは形容詞として使われており

  • not influenced

  • not ruled

  • not helped: not taking help or money from other people

という意味を持つ.これを見ただけでは progress に先行するのは意外な感じがする.類義語を調べてみると new, other, local, different, important, separate が現れる.

independent progressという組合せはめったに出てこないようだが,new progress とか important progress とかはなじんでいる感じがするので new or important くらいの意味だろう.

5文目

And now? The younger generation scarcely knows what Abelian functions are.”

scarcely

次はこの部分を調べる.

The younger generation scarcely knows what Abelian functions are.

scarcelyというのはここでは副詞として使われており,

  • almost not

  • certainly not

という意味を持つ.類義語として hardly, barely, not, never などがある.

語尾の -ly を取ると scarce という単語になり,これは not easy to find or get という意味である.

6文目

(From the modern viewpoint, the theory of Abelian functions is an analytic aspect of the theory of Abelian varieties, that is, projective algebraic group varieties; compare the historical sketch.)

和訳

The turn of the 20th century saw a sharp change in attitude to algebraic geometry.

20世紀に入ってから,ひとびとの代数幾何学への印象が急に変化した.

In the 1910s Klein writes as follows:

1910年代に,Klein が次のように書いている.

“In my student days, under the influence of the Jacobi tradition, Abelian functions were considered as the unarguable pinnacle of math.

『私の学生時代には,Jacobi からの伝統で Abel 関数の理論こそが疑いもなく数学の花形であるとされていた.

Every one of us felt the natural ambition to make some independent progress in this field.

私たちの誰もが,Abel 関数の理論において独自の結果をあげてやろうという自然な野心を抱いていたものだった.

And now? The younger generation scarcely knows what Abelian functions are.”

今はどうか?若いひとたちは Abel 関数の定義さえほとんど知らない』

(From the modern viewpoint, the theory of Abelian functions is an analytic aspect of the theory of Abelian varieties, that is, projective algebraic group varieties; compare the historical sketch.)

(現代的な観点からは,Abel 関数の理論は Abel 多様体の理論の解析的な側面であると位置づけられる.ここで Abel 多様体とは,射影代数群のことである.略史を参照のこと)

第3段落

Algebraic geometry had become set in a way of thinking too far removed from the set-theoretic and axiomatic spirit that determined the development of math at the time. It was to take several decades, during which the theories of topological, differentiable and complex manifolds, of general fields, and of ideals in sufficiently general rings were developed, before it became possible to construct algebraic geometry on the basis of the principles of set-theoretic math.

1文目

Algebraic geometry had become set in a way of thinking too far removed from the set-theoretic and axiomatic spirit that determined the development of math at the time.

the set-theoretic and axiomatic spirit

determined the development of math at the time を従えた that節は the set-theoretic and axiomatic spirit を修飾している.そこでいったん除外して

Algebraic geometry had become set in a way of thinking too far removed from the set-theoretic and axiomatic spirit.

という文章に還元する.

far removed from

far removed from something という組合せで to be very differnt from something という意味になる.したがって too far removed from the set-theoretic and axiomatic spirit 全体で a way of thinking を修飾しているものと思われる.

become set in

had become set は become set の過去完了形だと思われる.ここで become set in の使用例を調べると

  • By the time I was born he seems to have become set in his ways.

という用例がある.set in your ways で not liking change in your life という意味になるので,本文でも似た意味で使われているのではないかと思う.

become は be動詞のようなものなので,be set in your ways というイディオムの意味を調べれば参考になるだろう.これは to do the same things every day and to not want to change those habits という意味である.たとえば

  • As people get older, they often become set in their ways.

という例文を見つけることができた.

いずれにせよ本文での become set in には not liking change という意味が含まれているとみて間違いないだろう.

2文目

It was to take several decades, during which the theories of topological, differentiable and complex manifolds, of general fields, and of ideals in sufficiently general rings were developed, before it became possible to construct algebraic geometry on the basis of the principles of set-theoretic math.

the theories

途中に出てくる「the theories of topological, differentiable and complex manifolds, of general fields, and of ideals in sufficiently general rings」は具体的な数学の理論のことを指している.

during which

during which は期間を表す言葉を修飾する言い回しで,

  • The scene during which he died was highly symbolic.

などと使う.この場合 during which the theories ... were developed はおそらく several decades を修飾しているのだろう.いったん除外しても文は成立する.そこで

It was to take several decades before it became possible to construct algebraic geometry on the basis of the principles of set-theoretic math.

という文に簡約できる.

it is to take (time)

It was to take... となっているのは,次の例文と同様な意味に違いない.

  • It was to take several years before the outworks were completed.

つまり,意味の上では It took (a particular time) before... と同じであると考えられる.It was.. をはさむのは強調のためだろう.

和訳

Algebraic geometry had become set in a way of thinking too far removed from the set-theoretic and axiomatic spirit that determined the development of math at the time.

当時の数学の最先端の結果は,集合論に基づいた公理的な考え方を導入することで得られていたが,代数幾何学はそれとは異なる手法に固執していた.

It was to take several decades, during which the theories of topological, differentiable and complex manifolds, of general fields, and of ideals in sufficiently general rings were developed, before it became possible to construct algebraic geometry on the basis of the principles of set-theoretic math.

それから数十年の間に位相多様体や微分可能多様体,複素多様体の理論や抽象的な体論や環論が発達し,代数幾何学も集合論的な原理のもとに構成できるようになった.

第4段落

Towards the middle of the 20th century algebraic geometry had to a large extent been through such a reconstruction. Because of this, it could again claim the place it had once occupied in math. The domain of application of its ideas had grown tremendously, both in the direction of algebraic varieties over arbitrary fields and of more general complex manifolds. Many of the best achievements of algebraic geometry could be cleared of the accusation of incomprehensibility or lack of rigour.

1文目

Towards the middle of the 20th century algebraic geometry had to a large extent been through such a reconstruction.

to a large extent

have to a large extent という表現が気になる.調べてみると,次のような例文が見つかる.

  • Now the distinction has to a large extent disapperared.

もともと the distinction has disapperared という文章があって,それが to a large extent によって修飾されているようである.

そこで to a large extent を除くと,

Towards the middle of the 20th century algebraic geometry had been through such a reconstruction.

という文章が得られる.

towards

アメリカでは toward という言い方をし,イギリスでは towards という言い方をするらしい.

ここでは near to, just before, or around a time or place という意味で使われているようである.同じ意味で使われている例文として次がある.

  • I often get hungry towards the middle of the morning.

なお toward の -ward は語源的には forward や backward の -ward と同じである.

be through

例文を調べると,次のようなものが見つかる.

  • Alice had been through the medical doctor route.

  • The building has been through two more renovations since.

  • America has been through difficult times in the past.

be through を立項した辞書は見つからなかったが,おそらくこうした例文とそう違わない意味だろう.

2文目

Because of this, it could again claim the place it had once occupied in math.

it

it とは何だろうか.二つ登場するが,どちらもおそらく algebraic geometry のことだろう.because of this は,その前の文の内容を受けているものと思われる.

そこで,次の文に簡約される.

Algebraic geometry could again claim the place it had once occupied in math.

occupy

意味を調べると,次のようなことが載っている.

  • fill: to fill, exist in, or use a place or period of time

  • take control: to move into and take control and/or possession of a place

ここでは fill に近いと思われる.

a/the place を目的語に取る例としては,調べると次のようなものがある.

  • Salvation theory occupies a place of special significance in many religions.

ここでの意味と合致しているように見える.

3文目

The domain of application of its ideas had grown tremendously, both in the direction of algebraic varieties over arbitrary fields and of more general complex manifolds.

it

ここでも it は algebraic geometry を指すものと思われる.

the direction

both in the direction of… 以下は

  • algebraic varieties over arbitrary fields

  • more general complex manifolds

が並列されている.

4文目

Many of the best achievements of algebraic geometry could be cleared of the accusation of incomprehensibility or lack of rigour.

accusation

be cleared of the accusation の部分が述語になっており,

  • incomprehensibility

  • lack of rigor

が並列的に accusation を修飾しているように見える.

accusation は動詞 accuse が名詞化されたものである.

動詞 accuse は,調べたところ次のような意味を持つようである.

  • to say that you believe someone is guilty of a crime or of doing something bad

たとえば,次のように使う.

  • Smith accused her of lying.

  • He was accused of murder.

  • They accused him of cheating.

incomprehensibility

incomprehensible という形容詞があり,それの名詞形.

incomprehensible の意味は,difficult or impossible to understand である.

rigour

名詞であり,great care and thoroughness in making sure that something is correct という意味である.

  • Their research seems to me to be lacking in rigour.

などのように使う.

be cleared of

clear には(全部ではないが)次のような意味がある.

  • able to be understood

  • having no doubt, certain, or obvious

  • not marked or cloudly, pure, easy to see through

  • without being or feeling guilty

  • free from confusion

  • without problems or difficulties

be clear of で例文を調べると,次のようなものがヒットする.

  • Your campaign was clear of any such disasters.

be cleared of と be clear of は似たような意味と思われるが,be cleared of の方は後に続く言葉が charge や accusation などの crime っぽい言葉が多いように思われる.

和訳

Towards the middle of the 20th century algebraic geometry had to a large extent been through such a reconstruction.

20世紀の半ば,代数幾何学の大部分が(集合論の言葉によって)再構築された.

Because of this, it could again claim the place it had once occupied in math.

これにより,代数幾何学は再び数学においてかつての地位を取り戻したと言えるかもしれない.

The domain of application of its ideas had grown tremendously, both in the direction of algebraic varieties over arbitrary fields and of more general complex manifolds.

代数幾何の取り扱う範囲も大きく広がった.任意の体上の代数多様体が考えられるようになり,さらにより一般の複素多様体が考えられるようになった.

Many of the best achievements of algebraic geometry could be cleared of the accusation of incomprehensibility or lack of rigour.

かつて理解しづらく厳密性に欠けると言われていた数々の代数幾何の定理も,その汚名を返上できたのではないだろうか.

第5段落

The foundation for this reconstruction was algebra. In its first versions, the use of precise algebraic apparatus often led to a loss of the brilliant geometric style characteristic of the preceding period. However, the 1950s and 60s have brought substantial simplifications to the foundation of algebraic geometry, which have allowed us to come significantly closer to the ideal combination of logical transparency and geometric intuition.

1文目

The foundation for this reconstruction was algebra.

2文目

In its first versions, the use of precise algebraic apparatus often led to a loss of the brilliant geometric style characteristic of the preceding period.

apparatus

調べると,apparatus の意味は次の通り.

  • equipment: the set of tools and machines that you use for a particular scientific, medical, or technical purpose

  • machinery: the way in which a lot of people are organized to work together to do a job or control a company or country

characteristic

名詞として使われることが多いが,おそらくここでは形容詞.

いまでは typical of という言い方をすることが多いらしい.特に,

In its first versions, the use of precise algebraic apparatus often led to a loss of the brilliant geometric style typical of the preceding period.

と書き換えてもあまり意味は変わらない.typical of の例文として,ほかに

  • It features architectural elements typical of each city.

などがある.

typical of the preceding period 全体で the brilliant geometric style を修飾しているようなので,これを除外して

In its first versions, the use of precise algebraic apparatus often led to a loss of the brilliant geometric style.

として考えてよい.

3文目

However, the 1950s and 60s have brought substantial simplifications to the foundation of algebraic geometry, which have allowed us to come significantly closer to the ideal combination of logical transparency and geometric intuition.

substantial

substantial は次のような意味を持つ.

  • large: large in size, or importance

  • general: relating to the main or most important things being considered

和訳

The foundation for this reconstruction was algebra.

この代数幾何の再構築の基礎となったのは,代数学である.

In its first versions, the use of precise algebraic apparatus often led to a loss of the brilliant geometric style characteristic of the preceding period.

1回目の再構築では,代数的な道具を使ったことにより以前あった幾何学的な直観が失われてしまった.

However, the 1950s and 60s have brought substantial simplifications to the foundation of algebraic geometry, which have allowed us to come significantly closer to the ideal combination of logical transparency and geometric intuition.

しかし,1950年代と60年代の大規模な代数幾何学の基礎の単純化を経て,論理的な透明性と幾何学的直観が高い水準で両立できるようになった.

第6段落

The purpose of this book is to treat the foundations of algebraic geometry across a fairly wide front, giving an overall account of the subject, and preparing the ground for a study of the more specialised literature. No prior knowledge of algebraic geometry is assumed on the part of the reader, neither general theorems, nor concrete examples. Therefore along with development of the general theory, a lot of space is devoted to applications and particular cases, intended to motivate new ideas or new ways of formulating questions.

1文目

The purpose of this book is to treat the foundations of algebraic geometry across a fairly wide front, giving an overall account of the subject, and preparing the ground for a study of the more specialised literature.

across ... front

across a wide front という表現はあまり見かけない.調べると,なぜか戦争関連の表現が多く出てくる.

  • The attack of 16 August was carried out across a wide front.

across a front でもやはり戦争関連の表現が出てくる.

  • Gough planned a four-phase attack across a front of 8 miles.

どうやら attack するような気持ちで treat しているらしい…?

account

次のような意味を持つ.

  • description: a written or spoken description that says what happens in an event or process

  • at a bank:

  • with a shop/company

  • bill

  • arrangement to sell goods

ここでは description に近い意味だろう.

2文目

No prior knowledge of algebraic geometry is assumed on the part of the reader, neither general theorems, nor concrete examples.

assume

assume という言葉は think that something is true という意味で使われているが,assume on という言い方は辞書に載っていなかった.例文を探すと,

  • You can't just assume on logic that the shorter path is always better.

という文が見つかった.assume on という言い方をせずに

No prior knowledge of algebraic geometry is required to understand this book.

などとしてもほぼ同じ意味なのではないかと思う.

3文目

Therefore along with development of the general theory, a lot of space is devoted to applications and particular cases, intended to motivate new ideas or new ways of formulating questions.

和訳

The purpose of this book is to treat the foundations of algebraic geometry across a fairly wide front, giving an overall account of the subject, and preparing the ground for a study of the more specialised literature.

本書の目的は,代数幾何学の基礎を多角的に紹介することである.この分野全体の説明を行い,より専門的な研究に進むための基礎を身に着けてもらう.

No prior knowledge of algebraic geometry is assumed on the part of the reader, neither general theorems, nor concrete examples.

読者は代数幾何学に対する前提知識を持っていなくてもよい.定理も具体例も知らなくて大丈夫.

Therefore along with development of the general theory, a lot of space is devoted to applications and particular cases, intended to motivate new ideas or new ways of formulating questions.

そのため本書で一般論を展開するにあたり,新しい概念や問題意識を動機づけるための具体例の説明には紙数を割いた.

第7段落

It seems to me that, in the spirit of the biogenetic law, the student who repeats in miniature the evolution of algebraic geometry will grasp the logic of the subject more clearly. Thus, for example, the first section is concerned with very simple properties of algebraic plane curves. Similarly, Part I of the book considers only algebraic varieties in an ambient projective space, and the reader only meets schemes and the general notion of a variety in Part II.

1文目

It seems to me that, in the spirit of the biogenetic law, the student who repeats in miniature the evolution of algebraic geometry will grasp the logic of the subject more clearly.

biogenetic

biogenetic を持ち出しているのは不思議な感じがする.

miniature

miniature はここでは名詞で,in miniature で「exactly like something or someone but much smaller」という意味.

grasp

grasp は completely understand a fact or idea, especially a complicated one という意味で使われている.

残り

Thus, for example, the first section is concerned with very simple properties of algebraic plane curves. Similarly, Part I of the book considers only algebraic varieties in an ambient projective space, and the reader only meets schemes and the general notion of a variety in Part II.

ambient

ambient という形容詞は,existing in the surrounding area という意味で使われている.

和訳

It seems to me that, in the spirit of the biogenetic law, the student who repeats in miniature the evolution of algebraic geometry will grasp the logic of the subject more clearly.

これは私の持論だが,代数幾何の成長を追体験した学生は,この分野の考え方をよりよく理解できる.生物の発生を理解すれば,その種がより理解できるのと同じことだ.

Thus, for example, the first section is concerned with very simple properties of algebraic plane curves.

そこで最初のセクションでは平面代数曲線の簡単な性質を扱うことにした.

Similarly, Part I of the book considers only algebraic varieties in an ambient projective space, and the reader only meets schemes and the general notion of a variety in Part II.

同様に,本書の第1部では射影空間に埋め込まれているような代数多様体しか扱わない.読者がスキームや一般の多様体に触れるのは第2部以降である.

第8-9段落

Part III treats algebraic varieties over the complex number field, and their relation to complex analytic manifolds. This section assumes some acquaintance with basic topology and the theory of analytic functions.

1文目

Part III treats algebraic varieties over the complex number field, and their relation to complex analytic manifolds.

2文目

This section assumes some acquaintance with basic topology and the theory of analytic functions.

acquaintance

acquaintance には複数の意味があり,列挙すると次のようになる.

  • a person that you have met but do not know well

  • used in some expressions about knowing or meeting people

  • knowledge of a subject

ここでは単に knowledge という意味であろう.

和訳

Part III treats algebraic varieties over the complex number field, and their relation to complex analytic manifolds.

第3部では複素数体上の代数多様体を扱い,複素数体上の解析的な多様体との関連を議論する.

This section assumes some acquaintance with basic topology and the theory of analytic functions.

このセクションでは,予備知識として基礎的な位相幾何学と解析的関数についての理論の知識が仮定されている.

第9段落

第9段落は謝辞である.