I thought I’d give a list of results from enumerative geometry, as something I could refer back in the future. The reference is Eisenbud & Harris’s forthcoming book on enumerative geometry / intersection theory. I am nowhere ready to understand the whole machinery , but the results are perfectly understandable (in fact more understandable than a typical first course in algebraic geometry).

• There are 27 lines on a cubic surface in .
• Discriminant locus of deg- hypersurfaces in is a hypersurface of degree in , where (as always)
• Locus of deg- curves in having a triple point is of codimension 4 in , with degree

His Imperial Majesty, Tsai-Shun, deputed by Heaven to reign over all within the four seas, expired on the evening of Tuesday the 13th January 1875, aged eighteen years and nine months. He was erroneously known to foreigners as the Emperor T’ung Chih; but T’ung Chih was merely the style of his reign, adopted in order that the people should not profane by vulgar utterance a name they are not even permitted to write.[*] Until the new monarch, the late Emperor’s cousin, had been duly installed, no word of what had taken place was breathed beyond the walls of the palace; for dangerous thoughts might have arisen had it been known that the State was drifting rudderless, a prey to the wild waves of sedition and lawless outbreak. The accession of a child to reign under the style of Kuang Hsu was proclaimed before it was publicly made known that his predecessor had passed away.

[*] Either one or all of the characters composing an emperor’s name are altered by the addition or omission of certain component parts; as if, for instance, we were to write an Albart chain merely because Albert is the name of the heir-apparent. Similarly, a child will never utter or write its father’s name; and the names of Confucius and Mencius are forbidden to all alike.

Of the personal history of the ill-fated boy who has thus been prematurely cut off just as he was entering upon manhood and the actual government of four hundred million souls, we know next to nothing. His accession as an infant to the dignities of a sensual, dissipated father, attracted but little attention either in China or elsewhere; and from that date up to the year 1872, all we heard about His Majesty was, that he was making good progress in Manchu, or had hit the target three times out of ten shots at a distance of about twenty-five yards. He was taught to ride on horseback, though up to the day of his death he never took part in any great hunting expeditions, such as were frequently indulged in by earlier emperors of the present dynasty. He learnt to read and write Chinese, though what progress he had made in the study of the Classics was of course only known to his teachers. Painting may or may not have been an Imperial hobby; but it is quite certain that the drama received more perhaps than its full share of patronage. The ladies and eunuchs of the palace are notoriously fond of whiling away much of their monotonous existence in watching the grave antics of professional tragedians and laughing at the broad jokes of the low-comedy man, with his comic voice and funnily-painted face. Listening to the tunes prescribed by the Book of Ceremonies, and dining in solemn solitary grandeur off the eight[*] precious kinds of food set apart for the sovereign, his late Majesty passed his boyhood, until in 1872 he married the fair A-lu-te, and practically ascended the dragon throne of his ancestors. Up to that time the Empresses-Dowager, hidden behind a bamboo screen, had transacted business with the members of the Privy Council, signing all documents of State with the vermilion pencil for and on behalf of the young Emperor, but probably without even going through the formality of asking his assent. The marriage of the Emperor of China seemed to wake people up from their normal apathy, so that for a few months European eyes were actually directed towards the Flowery Land, and the Illustrated London News, with praiseworthy zeal, sent out a special correspondent, whose valuable contributions to that journal will be a record for ever. The ceremony, however, was hardly over before a bitter drop rose in the Imperial cup. Barbarians from beyond the sea came forward to claim the right of personal interview with the sovereign of all under Heaven. The story of the first audience is still fresh in our memories; the trivial difficulties introduced by obstructive statesmen at every stage of the proceedings, questions of etiquette and precedence raised at every turn, until finally the kotow was triumphantly rejected and five bows substituted in its stead. Every one saw the curt paragraph in thePeking Gazette, which notified that on such a day and at such an hour the foreign envoys had been admitted to an interview with the Emperor. We all laughed over the silly story so sedulously spread by the Chinese to every corner of the Empire, that our Minister’s knees had knocked together from terror when Phaeton-like he had obtained his dangerous request; that he fell down flat in the very presence, breaking all over into a profuse perspiration, and that the haughty prince who had acted as his conductor chid him for his want of course, bestowing upon him the contemptuous nickname of “chicken-feather.”

[*] These are–bears’ paws, deers’ tail, ducks’ tongues, torpedos’ roe, camels’ humps, monkeys’ lips, carps’ tails, and beef-marrow.

Subsequently, in the spring of 1874, the late Emperor made his great pilgrimage to worship at the tombs of his ancestors. He had previous to his marriage performed this filial duty once, but the mausoleum containing his father’s bones was not then completed, and the whole thing was conducted in a private, unostentatious manner. But on the last occasion great preparations were made and vast sums spent (on paper), that nothing might be wanting to render the spectacle as imposing as money could make it. Royalty was to be seen humbly performing the same hallowed rites which are demanded of every child, and which can under no circumstances be delegated to any other person as long as there is a son or a daughter living. The route along which His Majesty was to proceed was lined with closely-packed crowds of loyal subjects, eager to set eyes for once in their lives upon a being they are taught to regard as the incarnation of divinity; and when the Sacred Person really burst upon their view, the excitement was beyond description. Young and old, women and children, fell simultaneously upon their knees, and tears and sobs mingled with the blessings showered upon His Majesty by thousands of his simple-minded, affectionate people.

The next epoch in the life of this youthful monarch occurred a few months ago. The Son of Heaven[*] had not availed himself of western science to secure immunity from the most loathsome in the long category of diseases. He had not been vaccinated, in spite of the known prevalence of smallpox at Peking during the winter season. True, it is but a mild form of smallpox that is there common; but it is easy to imagine what a powerless victim was found in the person of a young prince enervated by perpetual cooping in the heart of a city, rarely permitted to leave the palace, and then only in a sedan-chair, called out of his bed at three o’clock every morning summer or winter, to transact business that must have had few charms for a boy, and possessed of no other means of amusement than such as he could derive from the society of his wife or concubines. Occasional bulletins announced that the disease was progressing favourably, and latterly it was signified that His Majesty was rapidly approaching a state of convalescence. His death, therefore, came both suddenly and unexpectedly; happily, at a time when China was unfettered by war or rebellion, and when all the energies of her statesmen could be employed in averting either one catastrophe or the other. For one hundred days the Court went into deep mourning, wearing capes of white fur with the hair outside over long white garments of various stuffs, lined also with white fur, but of a lighter kind than that of the capes. Mandarins of high rank use the skin of the white fox for the latter, but the ordinary official is content with the curly fleece of the snow-white Mongolian sheep. For one hundred days no male in the Empire might have his head shaved, and women were supposed to eschew for the same period all those gaudy head ornaments of which they are so inordinately fond. At the expiration of this time the Court mourning was changed to black, which colour, or at any rate something sombre, will be worn till the close of the year.

[*] Such terms as “Brother of the Sun and Moon” are altogether imaginary, and are quite unknown in China.

For twelve long months there may be no marrying or giving in marriage, that is among the official classes; the people are let off more easily, one hundred days being fixed upon as their limit. For a whole year it is illegal to renew the scrolls of red paper pasted on every door-post and inscribed with cherished maxims from the sacred books; except again for non-officials, whose penance is once more cut down to one hundred days’ duration. In these sad times the birth of a son–a Chinaman’s dearest wish on earth–elicits no congratulations from thronging friends; no red eggs are sent to the lucky parents, and no joyous feast is provided in return. Merrymaking of all kinds is forbidden to all classes for the full term of one year, and the familiar sound of the flute and the guitar is hushed in every household and in every street.[*] The ordinary Chinese visiting-card– a piece of red paper about six inches by three, inscribed with its owner’s name in large characters–changes to a dusky brown; and the very lines on letter paper, usually red, are printed of a dingy blue. Official seals are also universally stamped in blue instead of the vermilion or mauve otherwise used according to the rank of the holder. Red is absolutely tabooed; it is the emblem of mirth and joy, and the colour of every Chinese maiden’s wedding dress. It is an insult to write a letter to a friend or stranger on a piece of plain white paper with black ink. Etiquette requires that the columns should be divided by red lines; or, if not, that a tiny slip of red paper be pasted on in recognition of the form. For this reason it is that all stamps and seals in China arered–to enable tradesmen, officials, and others to use any kind of paper, whether it has already some red about it or not; and every foreigner in China would do well to exact on all occasions the same formalities from his employes as they would consider a matter of duty towards one of their own countrymen, however low he might be in the social scale.

[*] Mencius. Book v., part ii., ch. 4.

Certain classes of the people will suffer from the observance of these ceremonies far more severely than others. The peasant may not have his head shaved for one hundred days–inconvenient, no doubt, for him, but mild as compared with the fate of thousands of barbers who for three whole months will not know where to look to gain their daily rice. Yet there is a large section of the community much worse off than the barbers, and this comprises everybody connected in any way with the theatres. Their occupation is gone. For the space of one year neither public nor private performance is permitted. During that time actors are outcasts upon the face of the earth, and have no regular means of getting a livelihood. The lessees of theatres have most likely feathered their own nests sufficiently well to enable them to last out the prescribed term without serious inconvenience; but with us, actors are proverbially improvident, and even in frugal China they are no exception to the rule.

Officials in the provinces, besides conforming to the above customs in every detail, are further obliged on receipt of the “sad announcement” to mourn three times a-day for three days in a particular chapel devoted to that purpose. There they are supposed to call to mind the virtues of their late master, and more especially that act of grace which elevated each to the position he enjoys. Actual tears are expected as a slight return for the seal of office which has enabled its possessor to grow rich at the expense too often of a poor and struggling population. We fancy, however, that the mind of the mourner is more frequently occupied with thinking how many friends he can count among the Imperial censors than in dwelling upon the transcendent bounty of the deceased Emperor.

We sympathise with the bereaved mother who has lost her only child and the hope of China; but on the other hand if there is little room for congratulation, there is still less for regret. The nation has been deprived of its nominal head, a vapid youth of nineteen, who was content to lie perdu in his harem without making an effort to do a little governing on his own responsibility. During the ten years that foreigners have resided within half a mile of his own apartments in the palace at Peking, he has either betrayed no curiosity to learn anything at all about them, or has been wanting in resolution to carry out such a scheme as we can well imagine would have been devised by some of his bolder and more vigorous ancestors. And now once more the sceptre has passed into the hands of a child who will grow up, like the late Emperor, amid the intrigues of a Court composed of women and eunuchs, utterly unfit for anything like energetic government.

The splendid tomb which has been for the last twelve years in preparation to receive the Imperial coffin, but which, according to Chinese custom, may not be completed until death has actually taken place, will witness the last scene in the career of an unfortunate young man who could never have been an object of envy even to the meanest of his people, and who has not left one single monument behind him by which he will be remembered hereafter.

where the rows correspond to r=0 to 5. It’s sheer awesomeness that each number is twice the sum of the two numbers above it!

Two charges, with equal or opposite charge. The pencil of conics can be realized as a morphism

Note that the three polynomials are all bihomogeneous with bidegree (2,1). The affine curves on the xy-plane, i.e. on , are circles passing through x=±1,y=0, and on we have hyperbolae passing through y=0, z=±1.

This makes it explicitly a family of rational curves –– which generalizes to arbitrary charges at only those two points. This relates to the fact that rational curves (genus=0) of degree d≥3 in the plane necessarily are singular, with the difference of arithmetic and geometric genera equal to (d-1)(d-2)/2.

2. 保守力 conservative force。conservative源于conserve(守恒)。各种守恒定律都翻译的很好，偏偏conservative force无缘无故地称为保守。沾着一个守字，也算是差强人意吧。

Rational parametrization, as the name suggests, is to parametrize a variety by rational functions. The circle

can be parametrized by

with inverse

familiar even to high school students. In terms of homogeneous coordinates, we have a rational map given by

and the image is . [In general, any rational curve of degree in is given by homogeneous polynomials of degree in two variables, which may (if ) span the -dimensional vector space. Note that any conic is necessarily a plane conic, while we could get twisted cubics, twisted quartics, etc.]

The parametrization of the circle gives a 1-1 correspondence between the rational points on the unit circle and rational . From this, we obtain a complete description of the Pythagorean triples (treated in Euclid’s Elements).

What I really wanted to talk about, so as to force myself to work it out, is the simple pendulum problem. Let be the mass of the string, and be the length of the (massless) string. Let be the (generalized) coordinate, as measured from the vertical direction. Then,

The equation of motion is . In the so-called small angle approximation, we get , which is a harmonic oscillator. In the general case, we can’t solve it.

However, if we use the rational parametrization of the circle, i.e. use as the coordinate, we can get the Hamiltonian to be a rational function on two variables. Let’s see.

From , one finds that , and the Lagrangian becomes

The canonical momentum is

and the Hamiltonian is

The trajectories in the phase space, which are level sets of , are algebraic curves, which look like degree-6 curves… I was expecting some connection with elliptic curves though.

Something only occured to me today, that my earlier construction of the Torus with two families of Villarceau circles on it falls in the same sort of algebraic geometry. A torus is an algebraic variety of degree 4

on which there are 4 one-parameter families of circles. By a stretch of imagination (into the complex realm), any quartic surface should have 4 one-parameter families of conics on it (dim=1, with 4 components). On a quadric, there are lots of conics, one for each plane intersecting it. The parameter space is therefore , so it’s (dim=3, deg=1).

Further analogy can be drawn with the lines on a quadric. Any two lines from the same family don’t intersect, while each line in one family intersects any in the other family in a point. In the case of circles on the torus, any two circles from the same family are disjoint, whereas two from different families intersect in one or two points.

There must be some beautiful generalization of all these.

The parameter space of lines on a quadric is known as the Fano scheme , where the subscript 1 denotes the dimension of the subvarieties. A generalization of this would take into account the degree as well.

For a hypersurface  of degree , one expects its Fano scheme to be zero-dimensional, and the degree can be computed in Macaulay2:

On the other hand, Clemens’ Conjecture says that on a general quintic threefold in , there are finitely many rational curves of degree d for any given d. It is proven for d≤10, and the numbers are

曹公开篇即说，《红楼梦》中的故事，没有确切朝代，没有确切地点，但后辈考据一派非要争个子丑寅卯来。我原先也以为故事讲的是金陵的贾史王薛四大家族在北京的故事，或许是受近代红学的影响，或许不是。(记得小时候看电视里的智力问答说《红楼梦》用的是北京方言，当时还诧异了一下，可能就此认定是在北京了。) 书中的确常提“都中”，但从没明确说是哪里。南京的旧称“金陵”，“应天府”，“江宁”在书中屡次出现，而北京的历史名称一个也没有，这就不免有些奇怪。可以肯定的是“都”是南京以外的城市，就在南京城左近的说法过于牵强。林黛玉在进府之前都没怎么见过外祖母家的人（遂有所谓“宝黛初会”），可见“都”离扬州有一段距离。新版把它定在北京完全说得通，大可不必照曹公之意模模糊糊。如果上纲上线说是北京人自己往脸上贴金，那我也没什么好争竞的了。

有人还曾经考证《红楼梦》中的女子是大脚还是小脚，可谓无聊之极。南京北京之争还稍微有点意义，不过也多不到哪儿去。

一、国家制定的普通话标准发音自然是以北京语音为基础，但语音是随时间变化的，即便是近百余年由于科技的外因而使语音趋于恒定，但仍可以看出小小的差别。我把能想到的记录如下。

一、随时补充，仿明清笔记体例。

一、或有文白之分，以文转白，而非纯粹意义上的变音。1935年的《標準語大辭典》有“正音”、“變音”之分。

一、有些太口语的，属于念秃噜了的，就不录了。

一、千年以降，变音某种意义上说就是“误读”，几如通假字就是白字。（一说北京音为蒙满人学汉语没学好，夹杂口音。至今北京话许多词汇是“胡语”。）

一、总体来说，多音字的音趋向合并。是亦为一种“误读”。

一、京剧的念白以前都是韵白，为中州韵，古韵。后来的京白或保留一些民国时期的北京音。

一、找了本台湾近年（2003）出的《形音義規範字典》。两岸分歧之处或可一窥语音的演变。又找到一本1921年的《校改國音字典》，一本1935年初版的《標準語大辭典》，参考比较。大致上台湾沿袭民国时的大部分读法，而大陆由于进行过语言改革（从拼音到简体字、异体字），字音有的也重新规定了。

和

作为连词（或介词），表示“与”的意思，台湾直到现在都读作han4，其实早在二三十年代在北京就已经不流行了，统一读作he2（如“和平”）。只有在一些口语还保留，如“哪儿和哪儿”。

亦可作动词，如“和衣而卧”，“和盘托出”。

白

文言读作bo2，如李白，白居易的英文拼写一直都是Po，前几天我还看见白居易的汉语拼音注成Bo Juyi呢。民国时有些文人还习惯念bo2。京剧里的念白把“白虎堂”的白念作bo2。（但“白话”的白还应该读成bai2，以示为白话。）

又，“伯父”的伯就念bo2，但北京说“叔伯兄弟”shu1bai，大伯子bai3，而大伯bai1（土一点），老伯bo2（文一点）。

又，柏树，我们习惯念bai3，但台湾柏字只有一个发音，念bo2。柏林（Berlin）的译文看来是按老的念法，北京的柏林寺也要读bo2。至于张柏芝是念bo2还是bai3，那就难说了。人家名字本来按粤语起的，北京念什么也没处找理去。

又，百的正音也是bo2，如今都念bai3了，只有地名百色、人名百里奚有人还念bo2。

叔

《形》只有shu2一个音。北京话有说“五叔”shu2的，但现在大多数情况还是念shu1。

又，“淑女”也读shu2。

尉

太尉现在大家都念wei4，但京剧里的念白念作yu4，就像复姓尉迟。40年代的英文翻译“轻车都尉”还是 ch’ing-che-tu-yu 呢。

冲

人名“林冲”在京剧里念成上声chong3，林教头是不是要出来辩说一番？

学

我本以为只是土话念xiao2（如侯宝林的相声），但张学良自己叫自己zhang1 xiao2liang2，而且有的正式出版物上拼成Hsiao而非Hsueh。看来“文转白”也不是绝对的。《標》正變之分，而《形》無xiao2音

觉

正音是jue2，俗音jiao2（另有睡觉jiao4）。有一次翻到大觉寺1900年左右的外文注音为chiao，怀疑北京人当时就念jiao2。又，爱新觉罗英文翻译成Aisin Gioro，严重怀疑念jiao2。

女

说道爱新觉罗，想起女真翻译成Jurchen，有可能是念“汝”。“女”可以作为通假字，通汝。

大

大夫是念dai4fu还是da4fu1，现在的念法是前者为医生，后者为古官职。但京剧里念大王为dai4wang1，或是古韵。

期

《形》《標》通读qi2，而非qi1。除了作“一周年”讲是读ji1，如“期服”（一种丧服）。

企

《形》《標》通读qi4，而非qi3。如“企业”，“企图”，“企鹅”。

识

《形》《標》多数情况念shi4，而非shi2。如“识字”，“相识”。现在北京话说“知识”，“见识”，“认识”，“意识”还像去声（通常念作轻生），一般（尤其作动词）已经习惯念阳平。大陆字典已统一标shi2了。（念zhi4的情形两岸是一致的。）

凸

《形》《標》通读tu2，而北京现在习惯念tu1。

姣

《形》：jiao3，如姣妻，姣好

纪

《形》：作姓氏也念ji4，而非ji3。

三更

《形》：不作jing1，而作geng1。

贾

人名陆贾是念gu3，还是念jia3？只有问他本人。多半接近于前者。jia3只作姓，而《標》无此音。

（《形》：沽gu1与贾gu3，音近义同。当“买”的意思时，可以贾任何东西，而只能“沽酒”。又“沽名”不能作“贾名”。表示“卖”时，“余勇可贾”习惯上用贾，“待价而沽”习惯上用沽。）

慑

《形》：zhe2，而非she4。如“震慑”，“威慑”，“慑服”。

惊蛰

《形》：zhi2，我习惯念zhe2

惩

《形》：cheng2，不是cheng3

胜

《形》：“胜任”，“不胜枚举”，还有作姓时是读sheng1。当赢讲，或“胜地”，“引人入胜”时，读sheng4。

匀称

《形》：cheng4，不是chen4

署

《形》：除作官署讲时读shu3，其他如“署名”，“部署”，“署理”均读shu4。

又，“曙光”也读shu4。

蜗牛

台湾念gua1niu2，令人费解。或许受地方影响，如垃圾le4 se4。

Last one first. It’s conspicuous that there is a circle going through all three points. Actually that’s a component of a deg 3 curve, with the other component being the x-axis. (Note generically the curves are deg 4.) The one above also seems to have a reducible curve, but it’s hard to compute what it is.

By the standard algorithm, we obtain and for each of the three cases. Here , but we may take it to be at some point. Since all the curves pass through the three points, the polynomials are contained in the following ideals

and, in addition, since the curves self-intersect at (1,0), we have

  but