Max Noether

How to use this course

Max Noether is often remembered as Emmy Noether’s father. In his own time he was one of the leaders of algebraic geometry, the subject that studies curves and surfaces defined by polynomial equations. He insisted on algebraic proofs of results that Bernhard Riemann had reached by analysis, and he was among the first to treat geometric properties as invariant under birational transformations. The Italian school of Guido Castelnuovo, Federigo Enriques and Francesco Severi took up his ideas directly. His daughter’s ideal theory later gave them the foundations he had lacked.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements in modern language, proof sketches and exercises with folded solutions. William Fulton’s Algebraic Curves is the best companion for Lessons 04–06, and the books of Arbarello–Cornalba–Griffiths–Harris and Barth–Hulek–Peters–Van de Ven for Lessons 06–08.

Timeline

Mannheim, 1844–1865

Max Noether was born in Mannheim on 24 September 1844, the third of five children in a prosperous Jewish family. His grandfather, Elias Samuel, had founded a wholesale business in Bruchsal in 1797. When Baden’s Tolerance Edict of 1809 required Jewish families to take fixed surnames, the family chose Noether, and his father, born Hertz, became Hermann. The family firm, dealing in iron and hardware, remained in the family until 1937, when the Nazi state seized Jewish businesses.

Noether attended the Mannheim Gymnasium until 1858, when at fourteen he contracted polio. He could not walk for two years and was disabled for the rest of his life. He finished the school curriculum at home and then taught himself university mathematics from books. Before entering university he spent a period at the Mannheim observatory and published a short astronomical note, the first sign of the interest in physical science that took him to Heidelberg.

Heidelberg and the Clebsch school, 1865–1870

In 1865 Noether entered the University of Heidelberg. He studied there for three semesters, mainly theoretical physics with Gustav Kirchhoff, and learned mathematics from Jacob Lüroth. On 5 March 1868 he received his doctorate. Heidelberg then allowed a doctorate on the basis of an oral examination alone, without a dissertation; by tradition the candidate supplied the wine.

On Lüroth’s advice Noether went to Giessen to work with Alfred Clebsch and Paul Gordan. Clebsch was then the leader of German algebraic geometry. He combined Riemann’s theory of algebraic functions with the invariant theory of Arthur Cayley and James Joseph Sylvester, and he had founded, with Carl Neumann, the journal Mathematische Annalen. At Giessen Noether also met Alexander von Brill, a young lecturer who became his closest collaborator and lifelong friend. When Clebsch moved to Göttingen later in 1868, Noether followed him.

Around Clebsch gathered the generation that would shape German geometry, among them Brill, Noether, Gordan and Felix Klein. Noether absorbed two influences that defined his work. From Riemann, through Clebsch, came the idea that an algebraic curve has invariants, such as the genus, that survive birational change of coordinates. From Luigi Cremona came the study of the birational transformations themselves.

Habilitation and the move to Erlangen, 1870–1875

In 1870 Noether habilitated at Heidelberg with the thesis Über Flächen, welche Schaaren rationaler Curven besitzen (on surfaces carrying families of rational curves) and became a Privatdozent. The years from 1870 to 1874 were his most creative. He published his first work on the generation of plane Cremona transformations (1870), the reduction of plane curve singularities by quadratic transformations (1871), the \(AF+BG\) theorem (1873), and the great memoir with Brill on algebraic functions (1874). On 25 September 1874 he was promoted to extraordinary professor at Heidelberg.

In 1875 he moved to the University of Erlangen in Bavaria as extraordinary professor. Klein had just left Erlangen for Munich, and Gordan had arrived from Giessen in 1874. Noether and Gordan, both from the Clebsch school and both Jewish, became close friends and remained colleagues for the rest of their careers. Noether stayed at Erlangen for 46 years.

The fundamental theorem: \(AF+BG\)

For the general reader. Two curves in the plane meet in a finite set of points. Which other curves pass through all those points? An obvious family comes from combining the two equations, \(AF+BG=0\). Noether found the precise local conditions under which a curve through the intersection points must be of this form. The result, now called Noether’s fundamental theorem, is the ancestor of the ideal theory his daughter created.

Statement

Work over an algebraically closed field \(k\). Let \(F,G,H\in k[X,Y,Z]\) be homogeneous, with \(F\) and \(G\) having no common component. For \(P\in\mathbb P^2\) let \(\mathcal O_P\) be the local ring of \(\mathbb P^2\) at \(P\), and write \(F_*,G_*,H_*\) for the dehomogenized equations near \(P\). We say Noether’s conditions hold at \(P\) if \(H_*\in(F_*,G_*)\mathcal O_P\).

The proof in Fulton’s book uses the fact that \(Z\) is not a zero-divisor modulo \((F,G)\) when no intersection point lies on the line \(Z=0\), and the isomorphism \(k[x,y]/(f,g)\cong\prod_{P}\mathcal O_P/(f,g)\) for affine curves meeting in finitely many points. In modern terms the theorem says that the ideal \((F,G)\) is saturated and is the intersection of its primary components at the points of \(F\cap G\), exactly the kind of statement Emanuel Lasker and Emmy Noether later made general.

Criteria

Here \(I(P,\cdot)\) is the intersection multiplicity and \(m_P\) the multiplicity. Case (1) is the one most often used: if two curves meet in \(mn\) distinct points, any curve through all of them is \(AF+BG\).

Proof. Cubics through eight points in general position form a pencil, so a cubic \(C\) through eight of the points is \(\lambda C_1+\mu C_2\) and contains the ninth. (A careful argument shows the eight points impose independent conditions on cubics.)

Cremona transformations and singularities

For the general reader. A Cremona transformation is a change of coordinates in the plane given by rational functions, invertible except on a few curves. Noether studied what such maps do to curves. He claimed that all of them are built from a single quadratic map and ordinary projective changes of coordinates, and he showed that every singular point of a plane curve can be simplified by repeating that quadratic map.

The standard quadratic transformation

The map \(\sigma(X:Y:Z)=(YZ:XZ:XY)\) is birational and equal to its own inverse, since \(\sigma\circ\sigma(X:Y:Z)=(X^2YZ:XY^2Z:XYZ^2)=(X:Y:Z)\) off the triangle \(XYZ=0\). It blows up the three points \((1:0:0),(0:1:0),(0:0:1)\) and contracts the three lines joining them.

A Cremona transformation of degree \(n\) is given by a net of curves of degree \(n\) with base points of multiplicities \(m_1\ge m_2\ge\cdots\). Since general members meet in one moving point and have genus 0,

From these, Noether derived the inequality \(m_1+m_2+m_3\gt n\) for \(n\ge2\). Composing with a quadratic transformation centred at the three worst base points then lowers the degree to \(2n-m_1-m_2-m_3\lt n\), and induction finishes. The gap, noticed by Corrado Segre and others, is that the three points may not be in a position where a quadratic map can be centred, for example when some are infinitely near. Castelnuovo gave the first complete proof in 1901.

Resolving singularities of plane curves

Each quadratic transformation centred at a singular point separates its tangent directions, much as a blow-up does. A numerical invariant built from the multiplicities of all singular and infinitely near points decreases at each step. The result gives the genus formula for a plane curve of degree \(n\):

the sum running over all singular points, including infinitely near ones.

Brill–Noether: adjoints and Riemann–Roch

For the general reader. Riemann’s theory of algebraic functions rested on analysis and on a principle that Weierstrass had shown to be shaky. Brill and Noether set out to prove its main results using only algebra and geometry, with curves drawn through the singular points of a given curve. Their 1874 memoir opened the study of special linear series, a subject still active today.

Adjoints and the residue theorem

Let \(C\) be an irreducible plane curve of degree \(n\) with only ordinary singular points (Lesson 05). A curve \(G\) is adjoint to \(C\) if \(m_P(G)\ge m_P(C)-1\) at every singular point \(P\). Divisors on \(C\) are counted on its smooth model \(X\), with the points over each singular point counted separately.

The proof is an application of the \(AF+BG\) theorem, with criterion 3 of Lesson 04 handling the singular points. From the residue theorem, Brill and Noether derived Riemann–Roch by a counting argument, and showed that the adjoints of degree \(n-3\) cut out the canonical series, of dimension \(g-1\) and degree \(2g-2\).

Special series and the Brill–Noether number

A \(g^r_d\) is a linear series of degree \(d\) and dimension \(r\). It is special if \(\ell(K-D)\gt0\). Brill and Noether counted parameters to predict that on a general curve of genus \(g\), the \(g^r_d\) form a family of dimension

and that none exist when \(\rho\lt0\). The existence part was proved by Kempf and by Kleiman–Laksov (1972–74), and the non-existence and exact dimension by Phillip Griffiths and Joe Harris in 1980.

Canonical curves

For the general reader. Every curve of genus at least 3 that is not hyperelliptic has a preferred shape in space, determined by its holomorphic differentials. Noether proved in 1880 that this canonical model is as simple as possible algebraically: every polynomial function on the curve comes from a polynomial in the ambient space.

Let \(X\) be a smooth projective curve of genus \(g\ge2\) and \(\omega_1,\dots,\omega_g\) a basis of holomorphic differentials. The canonical map is \(\varphi_K:X\to\mathbb P^{g-1}\), \(P\mapsto(\omega_1(P):\cdots:\omega_g(P))\). It is an embedding exactly when \(X\) is not hyperelliptic, and the image, of degree \(2g-2\), is the canonical curve.

Since \(h^0(mK)=(2m-1)(g-1)\) for \(m\ge2\) by Riemann–Roch, the number of independent hypersurfaces of degree \(m\) containing the canonical curve is \(\binom{g-1+m}{m}-(2m-1)(g-1)\). Enriques and later Karl Petri strengthened the result: the ideal of the canonical curve is generated by quadrics, except for trigonal curves and plane quintics, where cubics are also needed. Mark Green’s conjecture on syzygies, proved for general curves by Claire Voisin in 2002–05, is the far-reaching extension of this line.

Algebraic surfaces

For the general reader. After curves, the next case is surfaces. Here birational maps can blow points up into curves, which makes classification much harder. Noether found invariants that survive such maps and a relation among them, Noether’s formula, and an inequality that limits which values they can take. The Italian school built the classification of surfaces on these foundations.

Invariants

For a smooth projective surface \(S\) over \(\mathbb C\), the geometric genus is \(p_g=h^0(S,K_S)\), the number of independent holomorphic 2-forms, and the irregularity is \(q=h^1(S,\mathcal O_S)\). Clebsch introduced \(p_g\) as the number of independent adjoint surfaces of degree \(n-4\) to a surface of degree \(n\) in \(\mathbb P^3\), the analogue of Lesson 06, and Noether showed in his first papers that it is a birational invariant. Cayley, Zeuthen and Noether also defined the arithmetic genus \(p_a\) by a formula; the difference \(q=p_g-p_a\) became the irregularity. Set \(\chi(\mathcal O_S)=1-q+p_g\).

Noether found the formula in a classical form, expressed through the genera and other projective characters of a surface. Today it is the case \(D=0\) of the Hirzebruch–Riemann–Roch theorem, and together with the Riemann–Roch theorem for surfaces, which Noether, Enriques and Castelnuovo developed between 1886 and 1897, it gives

Surfaces with \(K^2=2p_g-4\) lie on the “Noether line”; Eiji Horikawa classified them in the 1970s. Combined with the Bogomolov–Miyaoka–Yau inequality \(K^2\le 9\chi\), Noether’s inequality bounds the region where surfaces of general type can live.

Space curves

For the general reader. A curve in the plane is described by one equation. A curve in three-dimensional space usually needs more, and the relation between its degree, its genus and the surfaces through it is subtle. The Berlin Academy’s Steiner Prize of 1882 went jointly to Noether and Georges-Henri Halphen for independent memoirs on this problem.

Degree, genus and surfaces of low degree

Let \(C\subset\mathbb P^3\) be a smooth irreducible curve of degree \(d\) and genus \(g\), not contained in a plane. The central questions are which pairs \((d,g)\) occur and how the curves with given \((d,g)\) are organized. A natural approach, used in the classical work, is to ask for the least degree of a surface containing \(C\) and to count, by the methods of Lesson 06, how many independent surfaces of each degree pass through it.

Proof. By adjunction, \(K_C=(K_{\mathbb P^3}+aH+bH)|_C=(a+b-4)H|_C\), so \(2g-2=\deg K_C=(a+b-4)ab\).

The value \(\lfloor d^2/4\rfloor-d+1\) is in fact the largest genus of any smooth non-planar curve of degree \(d\) in \(\mathbb P^3\); Castelnuovo’s later bound for curves in \(\mathbb P^r\) generalizes it. Curves of maximal genus lie on quadrics. The finer question, which genera occur below the bound, was settled only by Laurent Gruson and Christian Peskine in the late 1970s.

Vanishing Hessians, with Gordan

For the general reader. Otto Hesse claimed in the 1850s that a polynomial whose matrix of second derivatives has identically zero determinant must secretly involve fewer variables. In 1876 Gordan and Noether showed that Hesse was right for up to four variables and wrong from five on. Their paper is one of the deepest in nineteenth-century invariant theory and still the starting point of a modern research area.

For a form \(f\in\mathbb C[x_0,\dots,x_n]\), the Hessian is \(\operatorname{hess}(f)=\det\bigl(\partial^2f/\partial x_i\partial x_j\bigr)\). If \(f\) is a cone, meaning that after a linear change of variables it does not depend on \(x_n\), then the last row of the Hessian matrix vanishes and \(\operatorname{hess}(f)=0\). Equivalently, \(f\) is a cone if and only if its partial derivatives \(\partial_0f,\dots,\partial_nf\) are linearly dependent.

The Hessian vanishes identically if and only if the partial derivatives are algebraically dependent: their Jacobian matrix is the Hessian matrix. So Hesse’s claim amounts to: algebraic dependence of the partials forces linear dependence.

Gordan and Noether also described all forms with vanishing Hessian in five variables. Careful modern proofs of their results have appeared only recently, which shows how intricate the original argument is.

Erlangen, 1875–1919

Erlangen was a small university town, and for decades its mathematics rested on two professors, Noether and Gordan. Noether waited thirteen years for a full professorship, which came on 16 April 1888. His teaching was careful and his manner reserved. Francis Macaulay later wrote that Noether mistrusted intuition and was never satisfied without an algebraic proof.

On 28 August 1880 he married Ida Amalia Kaufmann, from a wealthy Jewish merchant family in Cologne. They had four children: Emmy (1882), Alfred (1883), who studied chemistry and died in 1918, Fritz (1884), who became an applied mathematician, and Gustav Robert (1889), who was in poor health all his life and died in 1928. The family lived comfortably in Erlangen. Emmy studied there from 1904 under Gordan, a family friend, and after her doctorate in 1907 she stayed at the institute without pay, supervising students and substituting for her father when he was ill.

Editor, historian, teacher

From 1893 Noether served on the editorial board of Mathematische Annalen, which had published a major paper of his nearly every year since 1870. In 1894 he and Brill published a report of more than 450 pages in the Jahresbericht of the German Mathematical Society on the development of the theory of algebraic functions. Brill later credited Noether with most of it. Noether also wrote long obituaries of the leading mathematicians of his age, among them Cayley (1895), Sylvester (1898), Francesco Brioschi (1898), Sophus Lie (1900), Charles Hermite (1901), Cremona (1904), George Salmon (1905), Lüroth (1911), Gordan (1914) and Zeuthen (1921). Together they form a history of nineteenth-century geometry.

His doctoral students included William Fogg Osgood (1890), later a leading American analyst at Harvard; the world chess champion Emanuel Lasker (1900), whose 1905 theory of primary ideals Emmy Noether would generalize; and, as co-supervisor with Paul Hensel, the philosopher Hans Reichenbach (1916).

Last years and legacy, 1915–1921

Ida Noether died in 1915. That year Emmy left for Göttingen at the invitation of David Hilbert and Klein. Alfred died in 1918. Max Noether became emeritus in 1919 and continued writing to the end; his obituary of Zeuthen appeared in his final year. He died in Erlangen on 13 December 1921, aged 77. Brill wrote his obituary for the German Mathematical Society, and Macaulay one for the London Mathematical Society.

He did not live to see what followed. Fritz Noether lost his post under the Nazis and emigrated to the Soviet Union, where he taught at Tomsk; he was arrested in 1937 and shot in 1941. Emmy was dismissed in 1933 and died in the United States in 1935. In 1937 the family firm was taken from the Noethers.

Honours

He shared the Steiner Prize of the Berlin Academy with Halphen in 1882. He was a corresponding member of the academies in Munich (1887) and Berlin (1896), a member of the Göttingen Society of Sciences, and also of academies in Budapest, Copenhagen, Turin and the Accademia dei Lincei in Rome. The London Mathematical Society made him an honorary member in 1913.

Legacy

• Birational geometry: Klein called him the founder of the algebraic theory of birational transformations of surfaces. The Noether–Castelnuovo theorem remains the basic structure result for the Cremona group.
• Curves: Brill–Noether theory and Max Noether’s theorem on canonical curves are central in the modern theory of moduli of curves.
• Surfaces: Noether’s formula and Noether’s inequality frame the geography of surfaces of general type.
• Algebra: the \(AF+BG\) theorem led, through Lasker and Emmy Noether, to primary decomposition and modern commutative algebra.
• The Italian school: Castelnuovo, Enriques, Segre and Severi regarded him, with Clebsch and Cremona, as a founder of their subject.

Connections

People linked to Max Noether across this collection. A name without a link has no page here yet. See Max Noether in the network of all 337 lives →

Teachers and advisors

• Gustav Kirchhoff
• Jacob Lüroth 1865-1868

Students

• Emanuel Lasker 1900
• Hans Reichenbach 1916
• William Fogg Osgood 1890

Influenced by

• Bernhard Riemann
• Gustav Kirchhoff 1865-1868
• Alfred Clebsch 1868-1872
• Luigi Cremona

Influenced

• Federigo Enriques
• Guido Castelnuovo

Colleagues and collaborators

• Felix Klein
• Paul Gordan 1875-1912
• Alexander von Brill 1868-1921

Rivals

• Georges-Henri Halphen 1882

Family

• Emmy Noether
• Fritz Noether

Also mentioned in this life

• Arthur Cayley
• Richard Dedekind
• Leonhard Euler
• Charles Hermite
• Heinrich Hertz
• David Hilbert
• Sophus Lie
• Blaise Pascal
• Jakob Steiner
• Karl Weierstrass

Fields

• algebraic geometry 17 lives
• invariant theory 7 lives

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Max Noether, MacTutor History of Mathematics.
2. Max Noether, Wikipedia.
3. Noether, Max, Neue Deutsche Biographie 19 (1999).
4. Noether, Max, Complete Dictionary of Scientific Biography.
5. A. Brill, Max Noether, Jahresbericht der DMV 32 (1923).
6. F. S. Macaulay, Max Noether, Proc. London Math. Soc., early 1920s.
7. A. Dick, Emmy Noether 1882–1935, Birkhäuser, 1981 (for the family).

Mathematics

1. M. Noether, Über einen Satz aus der Theorie der algebraischen Functionen, Math. Ann. 6 (1873).
2. A. Brill and M. Noether, Über die algebraischen Functionen und ihre Anwendung in der Geometrie, Math. Ann. 7 (1874).
3. P. Gordan and M. Noether, Über die algebraischen Formen, deren Hesse’sche Determinante identisch verschwindet, Math. Ann. 10 (1876).
4. A. Brill and M. Noether, Die Entwicklung der Theorie der algebraischen Functionen in älterer und neuerer Zeit, Jahresbericht der DMV 3 (1894).
5. W. Fulton, Algebraic Curves, free online edition (Noether’s theorem, residue theorem, Riemann–Roch).
6. A. B. Coble, Cremona transformations and applications, Bull. AMS 28 (1922).
7. E. Arbarello, M. Cornalba, P. Griffiths, J. Harris, Geometry of Algebraic Curves I, Springer, 1985.
8. W. Barth, K. Hulek, C. Peters, A. Van de Ven, Compact Complex Surfaces, Springer, 2nd ed., 2004.
9. J. Watanabe and M. de Bondt, On the theory of Gordan–Noether on homogeneous forms with zero Hessian, arXiv:1703.07624.
10. Noether–Enriques theorem, Encyclopedia of Mathematics.