Life and work of the mathematician Max Noether (1844-1921)

Macaulay, Francis S.

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Francis S. Macaulay(1862 – 1937)MAX NOETHER.S. XXXVII – XLII ausProceedings of the London Mathematical Society. - Second series.Volume 21. - London, 1923Digitale Ausgabe erstellt von Gabriele Do¨rflingerUniversita¨tsbibliothek Heidelberg, 2010Heidelberger Texte zur MathematikgeschichteMax Noether passed away on December 13th, 1921, at Erlangen, wherehe had been Professor of Mathematics for nearly fifty years. He was born atMannheim on September 24th, 1844, and took his doctorate at Heidelberg in1868. He was a corresponding or honorary member of many learned societies,including the Academies of Berlin, Go¨ttingen, Munich, Budapest, Copenhagen,Turin, the Royale Accademia dei Lincei, and the Institut de France. He waselected an honorary member of the London Mathematical Society in 1913 at thesame time as Adolf Hurwitz, Paul Painleve´, Corrado Segre, and Woldemar Voigt.He seldom, if ever, visited England, but expressed his regret that he was unableto attend the International Congress of Mathematics at Cambridge in 1912, asbe had reached an age when he was averse from travelling.Although not so well known to fame out of his own country as some otherfamous mathematicians, he was deservedly held in the highest esteem, and afterthe death of Clebsch became the recognised leader of the algebraic-geometricschool in Germany. He brought himself early to the front by his celebratedand fundamental “Satz aus der Theorie der algebraischen Functionen” (Math.Annalen, Vol. 6), which he followed in the next volume with the classical memoir“Ueber die algebraischen Functionen und ihre Anwendung in der Geometric,”written in collaboration with A. Brill. This memoir formed the solid foundation,and gave rise to the name, of “Geometry on a Curve,” which has been developedand extended, especially in Italy, into a geometry of greater depth and powerthan anything preceding it. How remarkable these developments have been isclearly shown in H. F. Baker’s fascinating review “Some Advances in the Theoryof Algebraic Surfaces,” published in our Proceedings, Ser. 2, Vol. 12, pp. 1–40.Noether’s mind was naturally intuitive, but he mistrusted intuition, and wasapt to let anything suggested by it pass out of his thought. It seemed as ifthis cramped his powers to some extent in his later years. Be found perhaps inhis subject that intuition was often liable to mislead. He was of course nevercontent without algebraic or arithmetic proof, but had sometimes to be satisfiedwith an incomplete proof. Although naturally impatient he would take infinitepains to understand the thought of others, and to give them abundant help outof his own ample resources. There are many, including the writer of this note,who are grateful to him for his help. He had searching and peculiar methods ofhis own for testing the truth of things.His name will always be intimately associated with the greatest of mathe-matical journals, the Mathematische Annalen. Scarcely a year passed from theappearance of the 2nd volume in 1870 till the 83rd in 1921 in which he did notmake an important contribution to its pages. With the publication of the 42ndvolume in 1893 he formally joined the editorial staff, and an obituary notice atthe beginning of the 84th volume bears eloquent testimony to the great value ofthe services he rendered it. He contributed greatly to the advancement of mathe-matical science in three distinct ways: by the new and fruitful ideas contained inhis original researches, by the patient investigation and encouragement he gave2to other writers, and by his acutely critical and detailed historical work. The lastconsisted of the great work which he and A. Brill compiled in 1894, “Entwicklungder Theorie der algebraischen Functionen in a¨lterer und neuerer Zeit,” for thethird volume of the Jahresbericht der Deutschen Mathematiker Vereinigung ; andelaborate memorial surveys of the work of many famous mathematicians, begin-ning with Cayley and Sylvester and ending with Zeuthen, which he contributedto the pages of the Math. Ann. The “Entwicklung, etc.” may be regarded asthe completion and rounding off of his algebraic-geometric work, and in it theNoetherian Theorem comes fully into its own.Of Noether’s original (as distinguished from critical and historical) work themost interesting, because of its far reaching developments, is that which waspublished in Vols. 6 and 7 of the Math. Ann., to which reference has alreadybeen made. This work is still so little known in England, notwithstanding thatit is fully reproduced in Clebsch’s “Geometrie,” that there seems good reasonto give an account of what it contains. In his first communication to the Math.Ann. (Vol. 2, p. 293) he makes use of a theorem which, up till then, had beentaken for granted, viz. that a plane curve f which passed through all the pointsof intersection of two given curves φ, ψ had an equation of the formf ≡ Aφ+Bψ = 0.The real question was whether these purely geometrical conditions were sufficient,and he there gave a formal proof that they were sufficient in the case in whichall the points were simple and finitely separated. He was thus led to considerthe more general and fundamental problem as to what were the simplest andsufficient conditions which would require a curve f to take the form Aφ + Bψ,when φ, ψ had common multiple points with contact of any degree of complexity.To this question he gave the answer in Vol. 6, p. 351 (which had been previouslypublished in the Go¨ttinger Nachrichten, 1872, p. 490), in a form which sooncame to be known as Noether’s fundamental theorem. He recognised intuitivelythat the problem was one of approximation, and that the answer to it could beformulated thus:— The necessary and sufficient conditions that f must satisfy inorder that it may be of the form Aφ+Bψ, are that f should be capable of beingwritten in the form A′φ + B′ψ when the origin is moved in succession to eachand every point of intersection of φ, ψ; where A′, B′ are ordinary power serieswhich are, for each common point of φ, ψ, quite independent of what they maybe for all the other points. This approximation of f with A′φ + B′ψ will havebeen carried far enough when, on equating coefficients of the terms in f andA′φ + B′ψ of a sufficiently high degree, the undetermined coefficients of A′, B′cannot all be eliminated so as to give a new relation among the coefficients of falone, i.e. a relation which is independent of all the similar relations obtainablefrom the equating of coefficients of terms of all lower degrees. It will be noticedthat no conditions have to be satisfied at infinity.3To the proof of this he added a simple rider, of fundamental importance for itsapplications, that if φ, ψ have no common tangent at a common multiple point,of order i for φ and j for ψ, then all the relations for the coefficients of f alonecorresponding to that intersection of φ, ψ come from the equating of coefficientsup to degree i+j−2, so that it is sufficient (but of course not necessary) for f tohave a multiple point of order i+j−1 at that point. At a later time he extendedthis, the so-called “simple” case, to the case where φ, ψ have higher singularitiesat a common point, which can be resolved by repeated transformation of typex′1 : x′2 : x′3 =1x1:1x2:1x3into separate common multiple points coming under the “simple” case. Thoughmuch has been written about the theorem these are the only important additionsthat have been made to it, beyond its extension in various ways to surfacesand hyperspaces. Of these the Lasker-Noether (named by Lasker the Noether-Dedekind) theorem is the most general and fundamental (Math. Ann., Vol. 60,p. 51; and Cambridge Tracts in Maths., etc., No. 19, p. 61).Brill and Noether’s classical memoir “Ueber die algebraischen Functionen,etc.,” in Vol. 7 of the Math. Ann., is founded essentially on the investigationof the properties of point-groups g(r)R on a given curve f = 0 cut out by a linearsystem of curves λ0φ0 + λ1φ1 + . . . + λrφr = 0, where φ0, φ1, . . . , φr are givenlinearly independent curves of the same order. Any fixed points common to f ,φ0, φ1, . . . , φr are left out of account, and R denotes the number of simple pointsin each group, while r is the freedom of the linear series g(r)R , i.e. the number ofthe R points which can be chosen at will to determine a particular group GR ofthe series g(r)R . Thus the curve f of order n can be birationally transformed to acurve F of order R by means of the transformation equationsx′1 : x′2 : x′3 = φ0 : φ1 : φ2;and this is possible so long as r ≥ 2. (The only exceptions are curves for whichthe genus p = 0, 1, 2, which require separate treatment; and hyperelliptic curves,or curves on which exist a series g(1)2 , since in this case the transformation isnot birational.) Hence one important problem is to find the least value of R forwhich a series g(2)R exists, as this value of R gives the lowest value of the order ofa curve F into which f can be birationally transformed.It was shown that the series g(r)R on f which were of the greatest interest andimportance were those which were cut out by a linear system (φ0, φ1, . . . , φr)adjoined to f , especially when the curves φ0, φ1, . . . , φrwere of order n − 3. Acurve φ adjoined to f is a curve having a multiple point of order i − 1 at eachordinary multiple point of order i on f , and a corresponding higher singularityat each point of higher singularity on f . If GR is any given group of R points onf , the series g(r)R of greatest freedom r, having GR as one of its groups, can be4found by drawing any adjoined curve through GR to cut f again in a group GQ(apart from the multiple points of f). Then the whole system of adjoined curvesthrough GQ of the order of φ will cut out on f the required series g(r)R of greatestfreedom. This series g(r)R is a fixed series, independent of the particular groupGQ by means of which it is derived, i.e. it is a series uniquely dependent onany one of its groups GR. This is one way of stating the Brill-Noether theoremof residuation, and, is proved by the help of the rider to Noether’s theoremmentioned above. This series g(r)R is called a complete coresidual series (everyone of its groups being residual to any group GQ to which any one of its groupsGR is residual) or the linear series of unrestricted freedom which any given groupGR determines.We will suppose now that any series denoted by g(r)R is a series of unrestrictedfreedom r unless the contrary is stated. In any group GR of g(r)R r points canbe chosen freely, and the remaining R− r points are dependent on the r points.Also R − r ≤ p, where p is the genus of the curve f . Further, if R − r < p,the group GR lies on an adjoined curve of order n − 3. (This is still more trueif the freedom r is restricted, for, on removing the restriction, r would increaseand R − r diminish ; but with restricted freedom we may have R − r ≥ p andyet GR on an adjoined curve of order n − 3.) Now, taking R − r < p, anddrawing all the adjoined (n-3)-ics which pass through a fixed group GR of g(r)R, we determine the series g(q)Q of freedom q, which is residual to the series g(r)R, where Q + R = 2p − 2. The two mutually residual series g(q)Q , g(r)R may beconsidered together; or g(r)R may be considered alone, in which case q must beregarded as a number not less intimately associated with the series g(r)R than ris; for r is the freedom of the series g(r)R on f , while q is the freedom of the wholesystem of adjoined (n-3)-ics through any fixed group GR of g(r)R . The numbersR, r, q, p are connected by the relationR = p− q + r − 1.This theorem is called the Riemann-Roch theorem for plane curves, and is alsoknown under the more definitive title of the Brill-Noether reciprocity theorem.Other ways of stating the theorem are(i) Q−R = 2(q − r), Q+R = 2(p− 1),and(ii) R− 2r = Q− 2q = p− q − r − 1 ≥ 0.In seeking the number of relations that must exist between the points of agroup GR on f , in order that the unrestricted series g(r)R determined bv GR maybe of freedom r, Brill and Noether obtained the formula (q+1)r, or r(p+r−R).5This was got by counting equations of condition, and is only valid under stringentlimitations, as also the inequalityp ≥ (q + 1)(r + 1)derived from it. It is possibly correct for curves f of given genus which are generalof their kind, and in the memoir it is only applied to such curves, but it requiresa strict proof. Assuming the correctness of the formula, then, in fixing the groupGR on f there are r(p+ r−R) points dependent on the rest by the equations ofcondition and R−r(p+r−R) which can be chosen freely. This cannot fall belowr, since a group of g(r)R has r free points even after GR has been fixed. HenceR− r(p+ r −R) ≥ r.This inequality determines the least value R can have for a given value of r ina series g(r)R . When r = 2, it becomes R ≥ 23(p+ 3). Hence a general curve f ofgiven genus p can be birationally transformed into a curve F of order R equal tothe integral part of 23(p+ 4). A curve of less order with given genus p, such as aquintic with p = 5, 6 or an n-ic with less than 12(n− 2)(n− 4) double points, isnot a general curve of genus p.The above result is not only interesting, and even surprising, in itself, butgains additional interest from its relation to Riemann’s theorem that a curve fof genus p (general of its kind) can be birationally transformed to a curve Fwhose coefficients depend on 3p− 3 parameters or moduli. A curve f of order nand genus p has 12n(n + 3) coefficients, which can be reduced to 12n(n + 3) − 8by a collinear transformation; these satisfy a number of independent relationsequal to the number of the double points, viz. 12(n − 1)(n − 2) − p, leaving3n+ p− 9 effective constants for f . The curve f can be transformed to a curveF of order R as above, and the process of transformation admits of imposing 0,1, 2 additional geometrical conditions on F (such as requiring it to have 0, 1, 2points of undulation) according as p = 0, 1, 2 mod 3. The number of effectiveconstants left for F are therefore 3R+p−9, 3R+p−10, 3R+p−11, i.e. 3p−3in all three cases, since 3R = 2p+6, 2p+7, 2p+8 when p = 0, 1, 2 mod 3. Thisreally only shows that the number of moduli can be brought as low as 3p − 3,not that it cannot be brought lower; but other proofs of the result are given inthe memoir.We have dwelt at length on these two memoirs because of their fundamentaland historical importance. The greater part of Noether’s original work may besaid to have been a natural continuation of them. He contributed much besidesto the theory of elimination, higher singularities, the geometry of hyperspace,and Abelian and Theta functions.F. S. M.6

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