Oscar Zariski

How to use this course

Oscar Zariski learned algebraic geometry from the great Italian masters, whose intuition was superb and whose proofs were often incomplete. In his late thirties he decided that the subject had to be rebuilt on the commutative algebra of Emmy Noether and Wolfgang Krull. Over the next fifteen years he did much of that rebuilding himself, and then trained a generation of students at Harvard who carried it into the age of schemes.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. The lessons follow the order of Zariski’s work: topology of plane curves (around 1929–1937), the algebraic foundations of 1937–1947, and the theory of holomorphic functions (1948–1951). Throughout, \(k\) denotes a field, algebraically closed unless stated otherwise.

Timeline

Kobrin and Kiev, 1899–1920

Ascher Zaritsky was born on 24 April 1899 in Kobrin, a small town in the western part of the Russian Empire, now Kobryn in Belarus. His parents, Bezalel Zaritsky and Hanna Tennenbaum, were Jewish. His mother made sure that he was well educated. A private tutor began teaching him at seven, and he was quick at arithmetic and at Russian.

The First World War brought the front line close to Kobrin, and the family moved east to Chernigov in Ukraine. In 1918 he went to the University of Kiev. The mathematics faculty was full, so he enrolled in philosophy and studied algebra and number theory on his own, through the years of revolution and civil war.

In 1920 he left for Italy, home of the most famous school of algebraic geometry in the world.

Rome, 1920–1927

In Rome he found three great geometers: Guido Castelnuovo, Federigo Enriques and Francesco Severi. Together they had built the classification of algebraic surfaces, using a style of reasoning that relied heavily on geometric intuition about linear systems of curves, degenerations and “generic” points. Zariski attended their lectures and was quickly drawn in. Castelnuovo became his mentor.

His thesis, suggested by Castelnuovo, was on a problem in Galois theory: which algebraic equations depending on a parameter can be solved by radicals. The question joins Galois groups to the geometry of curves, a mixture that foreshadowed his later work. He received his doctorate in 1924. In the same year he married Yole Cagli, whom he had met in Rome; the wedding took place in Kobrin, and their son Raphael was born in Rome in 1925 and their daughter Vera in Baltimore in 1932.

Enriques suggested an Italian form of his name, and his first paper, written with Enriques, appeared under the name Oscar Zariski. He held a fellowship of the International Education Board in Rome until 1927. Castelnuovo, who saw how algebraic his instincts were, reportedly told him “You are here with us but are not one of us.” The rise of Fascism made Italy increasingly dangerous for a foreign Jew, and in 1927 he accepted an offer from America.

Baltimore and the lost paradise, 1927–1937

With the help of Solomon Lefschetz, then at Princeton, Zariski went to the Johns Hopkins University in 1927 as a Johnston Scholar. His family joined him in 1928, and in 1929 he became a member of the faculty. For the next decade he made frequent trips to Princeton to talk with Lefschetz, whose topological approach to algebraic varieties became his first research programme in America. His work on the fundamental groups of complements of plane curves (Lesson 04) dates from these years.

In 1935 he published Algebraic Surfaces, a survey of the Italian theory for Springer’s Ergebnisse series. Writing it changed his life. Checking the arguments one by one, he found that many of the central proofs were incomplete or wrong, even when the theorems were true. In his own words this was the loss of “the geometric paradise in which I so happily had been living.”

He resolved to rebuild the foundations with algebra. He spent 1935–36 at the Institute for Advanced Study in Princeton and met regularly with Emmy Noether, then at Bryn Mawr, to learn the abstract algebra of ideals that his Italian teachers had never shown him. He studied Wolfgang Krull’s theory of valuations and local rings and Bartel van der Waerden’s first algebraic treatments of geometry. In 1937 he became a full professor at Johns Hopkins, and he was ready to begin the work described in Part II.

Fundamental groups of plane curves

For the general reader. Remove a curve from the complex projective plane and study the loops in what is left. The group of those loops records how the curve sits in the plane. Zariski discovered that two curves of the same degree with the same singularities can sit in the plane in genuinely different ways, and that the difference is controlled by a subtle piece of geometry: whether six special points lie on a conic.

The first homology

Let \(C\subset\mathbb P^2(\mathbb C)\) be a plane curve with irreducible components \(C_1,\dots,C_r\) of degrees \(d_1,\dots,d_r\). Write \(G=\pi_1(\mathbb P^2\setminus C)\). The abelianization of \(G\) depends only on the degrees.

This follows from Lefschetz duality, \(H_1(\mathbb P^2\setminus C)\cong H^3(\mathbb P^2,C)\), and the exact sequence of the pair, using that the fundamental class of \(C_i\) is \(d_i\) times the class of a line in \(H_2(\mathbb P^2)\cong\mathbb Z\). For a smooth curve Zariski showed that \(G\) itself is \(\mathbb Z/d\). He conjectured that the same holds for curves with only nodes; this was proved around 1980 by William Fulton and Pierre Deligne.

The sextic with six cusps

The two kinds of sextic have the same degree and the same singularities, yet their complements are not homeomorphic. Such a pair is now called a Zariski pair. Six points in general position lie on no conic, so the conic condition is a special position of the cusps. Later invariants also see the difference: the Alexander polynomial of the curve is \(t^2-t+1\) when the cusps lie on a conic and \(1\) otherwise.

Why the conic produces a free product

If the cusps lie on the conic \(f_2=0\), one can show that the sextic has an equation \(f_2^3+f_3^2=0\) with \(f_3\) a cubic. Consider the pencil of sextics \(\lambda f_2^3+\mu f_3^2\). The rational map \(\phi=[f_2^3:f_3^2]:\mathbb P^2\dashrightarrow\mathbb P^1\) sends \(C\) to the point \([1:-1]\). Its fibre over \([0:1]\) is the conic counted three times, and its fibre over \([1:0]\) is the cubic counted twice. Away from the six base points, \(\phi\) induces a surjection of \(G\) onto the orbifold fundamental group of \(\mathbb P^1\) minus one point, with cone points of orders 3 and 2, which is \(\mathbb Z/2*\mathbb Z/3\cong\mathrm{PSL}_2(\mathbb Z)\). Zariski’s computation shows that \(G\) is exactly this group. His technique of projecting from a point and following how the curve braids over the lines of the pencil was systematized by Egbert van Kampen in 1933 and is now called the Zariski–van Kampen method.

He later proved a hyperplane-section theorem for fundamental groups: for a hypersurface complement in \(\mathbb P^n\), the fundamental group equals that of its intersection with a generic plane. This reduces many questions to plane curves and is the homotopy analogue of the Lefschetz hyperplane theorem.

The Zariski topology

For the general reader. Over the complex numbers a variety has an ordinary topology. Over an arbitrary field it has none. Zariski’s idea was to declare the closed sets to be exactly the sets cut out by polynomial equations. The resulting topology is very coarse, but it is defined over any field and it captures exactly the geometry that algebra can see.

Definition

For a set \(S\subseteq k[x_1,\dots,x_n]\) let \(V(S)=\{a\in k^n: f(a)=0\ \forall f\in S\}\). For \(X\subseteq k^n\) let \(I(X)=\{f: f|_X=0\}\).

Zariski introduced the topology in the 1940s in his work on the Riemann manifold of a function field (Lesson 06), where its quasi-compactness was the key point. In 1960 Alexander Grothendieck extended it to the set \(\operatorname{Spec}R\) of all prime ideals of any commutative ring, with closed sets \(V(I)=\{\mathfrak p\supseteq I\}\). Every scheme carries this topology.

Irreducible components

A space is irreducible if it is not the union of two proper closed subsets, and Noetherian if closed sets satisfy the descending chain condition. By Hilbert’s basis theorem, \(\mathbb A^n\) with the Zariski topology is Noetherian.

The proof is Noetherian induction: a minimal closed set without such a decomposition would be reducible, and its two proper pieces would have decompositions. On the algebra side this is the Lasker–Noether theorem applied to radical ideals. The topology is far from Hausdorff: on an irreducible variety any two non-empty open sets meet, so open sets are dense.

Valuations and the Riemann manifold

For the general reader. A singular point of a surface can be approached along many different paths, and a valuation records one way of approaching it. Zariski collected all valuations of a function field into a single space. He proved that singularities can be removed near each valuation separately, and that the space is compact, so finitely many local repairs suffice. This was the strategy of his resolution theorems.

Valuation rings

Let \(K\) be a finitely generated field extension of \(k\), the function field of a variety. A valuation ring of \(K/k\) is a subring \(k\subseteq V\subseteq K\) such that for every \(x\in K^\times\), either \(x\in V\) or \(x^{-1}\in V\). Equivalently, \(V=\{x: v(x)\ge0\}\) for a valuation \(v:K^\times\to\Gamma\) into an ordered abelian group. If \(X\) is a variety with function field \(K\), the centre of \(V\) on \(X\) is the point (or irreducible subvariety) \(P\) such that \(\mathcal O_{X,P}\subseteq V\) and \(\mathfrak m_P=\mathfrak m_V\cap\mathcal O_{X,P}\). On a projective model the centre exists and is unique, by the valuative criterion of properness.

Zariski called this space the Riemann manifold of \(K\), since for a curve over \(\mathbb C\) it is the compact Riemann surface. It is now called the Zariski–Riemann space. It is the inverse limit of all projective models of \(K\). Local uniformization says each valuation has a neighbourhood in this space where some model is nonsingular. Compactness gives finitely many models \(X'_1,\dots,X'_s\), each nonsingular at the centres of an open set of valuations, together covering everything. The remaining hard step is to glue these finitely many partial resolutions into one, which Zariski accomplished in dimension 2 and 3 (Lesson 08).

The proof of compactness is close to the proof of Tychonoff’s theorem. Each valuation ring is determined by which elements of \(K\) it contains, so \(\mathrm{Zar}(K/k)\) embeds in a product of two-point spaces \(\{x\in V,\ x\notin V\}\) indexed by \(K\).

Normality and the Main Theorem

For the general reader. Some singular points look like several smooth sheets glued together; others are genuinely tangled. Zariski singled out the “normal” points, defined by an algebraic condition, and showed that a normal point cannot be partly blown up: under a birational change of model it either stays a single point or turns into something of positive dimension. This is Zariski’s Main Theorem, and its many reformulations run through modern algebraic geometry.

Normal points

A point \(P\) of a variety \(Y\) is normal if the local ring \(\mathcal O_{Y,P}\) is an integrally closed domain. A variety is normal if all its points are. Smooth points are normal, and in dimension 1 normal means smooth. The integral closure of the coordinate ring of an affine variety in its fraction field is a finitely generated algebra (Emmy Noether), so every variety \(Y\) has a normalization \(\tilde Y\to Y\), finite and birational, with \(\tilde Y\) normal. Zariski showed that the singular locus of a normal variety has codimension at least 2.

Grothendieck generalized this to a structure theorem: every quasi-finite separated morphism of finite presentation to a quasi-compact separated scheme factors as an open immersion followed by a finite morphism.

The finite case

Suppose \(f:X\to Y\) is finite and birational, with \(Y=\operatorname{Spec}A\) affine and normal and \(X=\operatorname{Spec}B\). Then \(A\subseteq B\subseteq\operatorname{Frac}A\) and \(B\) is a finite \(A\)-module, so every element of \(B\) is integral over \(A\). Since \(A\) is integrally closed, \(B=A\) and \(f\) is an isomorphism. The general case reduces to this one by Grothendieck’s factorization, or in Zariski’s original approach by a careful study of the local rings and valuations dominating \(\mathcal O_{Y,P}\).

Resolution of singularities

For the general reader. A singular variety has points where it fails to look smooth: crossings, cusps, pinches. Resolution of singularities asks for a smooth variety that maps onto it and differs from it only over the bad points. The Italians believed they had proved this for surfaces. Zariski gave the first algebraic proofs, for surfaces in 1939 and for three-dimensional varieties in 1944, and his student Heisuke Hironaka completed the theory in characteristic 0 in 1964.

Blowing up a point

The blow-up of \(\mathbb A^2\) at the origin is \(\mathrm{Bl}=\{((x,y),[u:v]): xv=yu\}\subset\mathbb A^2\times\mathbb P^1\). It is covered by two affine planes: the chart \(u\ne0\) with coordinates \((x,t)\), \(y=xt\), and the chart \(v\ne0\) with coordinates \((s,y)\), \(x=sy\). The preimage of the origin is the exceptional line \(E\cong\mathbb P^1\), given by \(x=0\) in the first chart. For a curve \(C=V(f)\) with a point of multiplicity \(m\) at the origin, \(f(x,xt)=x^m\tilde f(x,t)\), and \(V(\tilde f)\) is the strict transform.

For surfaces over \(\mathbb C\), Robert Walker had given a rigorous proof by analytic methods in 1935. Zariski’s algebraic method alternates two operations: normalize the surface, which removes singular curves and leaves isolated singular points (Lesson 07), then blow up those points. The difficulty is to prove that the process stops. Zariski did this through local uniformization of all valuations and the compactness of the Riemann manifold (Lesson 06), followed by a delicate gluing argument. In 1956 his student Shreeram Abhyankar resolved surfaces in positive characteristic. In positive characteristic and dimension at least 4, resolution remains open.

Simple points and regular rings

For the general reader. What does it mean for a point to be smooth? Over the complex numbers, the surface should look like a flat disc nearby. Over other fields the answer is less obvious. In 1947 Zariski compared the geometric definition, through derivatives, with Krull’s algebraic notion of a regular local ring, and found that they agree over perfect fields but can differ over imperfect ones.

Two definitions

Let \(X=V(f_1,\dots,f_m)\subseteq\mathbb A^n_k\) be irreducible of dimension \(d\), with \(I(X)=(f_1,\dots,f_m)\). A point \(P\) is simple (in Zariski’s terminology) if the Jacobian matrix \((\partial f_i/\partial x_j)(P)\) has rank \(n-d\). A Noetherian local ring \((R,\mathfrak m)\) is regular if \(\dim_{R/\mathfrak m}\mathfrak m/\mathfrak m^2=\dim R\). In general \(\dim\mathfrak m/\mathfrak m^2\ge\dim R\).

Idea. At a \(k\)-rational point \(P=0\), the Zariski tangent space \((\mathfrak m/\mathfrak m^2)^*\) is the kernel of the Jacobian matrix, because \(\mathfrak m/\mathfrak m^2\) is spanned by \(x_1,\dots,x_n\) modulo the linear parts of the \(f_i\). So \(\dim\mathfrak m/\mathfrak m^2=n-\operatorname{rank}J(P)\), and regularity is equivalent to rank \(n-d\). At non-rational points over imperfect fields, the residue field is inseparable over \(k\) and this computation breaks down. Zariski’s analysis made regular local rings the standard algebraic notion of nonsingularity, which Grothendieck adopted for schemes; smoothness of morphisms became the relative, geometric notion.

Formal functions and connectedness

For the general reader. Complex geometers study a variety near a subvariety using convergent power series. Over an arbitrary field there is no convergence, so Zariski used formal power series: completions of rings with respect to powers of an ideal. With them he proved that blowing up a normal point always produces a connected set, and that normal points stay irreducible even under infinitesimal magnification.

Holomorphic functions along a subvariety

Let \(f:X\to Y\) be a proper morphism of Noetherian schemes, \(y\in Y\), and \(X_n=X\times_Y\operatorname{Spec}(\mathcal O_{Y,y}/\mathfrak m_y^{n})\) the thickened fibres. Zariski’s holomorphic functions along the fibre are, in this language, compatible systems of functions on the \(X_n\). Grothendieck later extended his result to higher cohomology.

Proof sketch. Since \(Y\) is normal and \(f\) birational and proper, \(f_*\mathcal O_X=\mathcal O_Y\): sections of \(f_*\mathcal O_X\) are rational functions on \(Y\) that are integral over \(\mathcal O_Y\) (they form a finite module), hence regular. By formal functions, \(\widehat{\mathcal O}_{Y,y}\cong\varprojlim H^0(X_n,\mathcal O_{X_n})\). If \(f^{-1}(y)\) were a disjoint union of two non-empty closed sets, each \(X_n\) would split likewise, and the function equal to 1 on one piece and 0 on the other would give a non-trivial idempotent in the local ring \(\widehat{\mathcal O}_{Y,y}\), which is impossible. The Main Theorem follows quickly, since an isolated point of a connected fibre is the whole fibre.

Analytic irreducibility

A point is analytically irreducible if the completion of its local ring is a domain. Zariski proved in 1948 that normal points of varieties are analytically irreducible, and later that the completion of a normal local ring of a variety is again normal. Irreducible points need not be analytically irreducible, as the next exercise shows.

Harvard, 1947–1969

The war years at Johns Hopkins were the most productive of Zariski’s life. He was elected to the National Academy of Sciences in 1944 and received the Cole Prize in Algebra the same year. In 1945 he spent a year at the University of São Paulo, where André Weil, then teaching there, was the most important member of his audience; the two had already corresponded intensively about foundations, and Weil later said Zariski’s were the only works in algebraic geometry he fully trusted. After a year at the University of Illinois, Zariski moved to Harvard in 1947. He was the first Jewish member of its mathematics department, served as chairman, and stayed for the rest of his life.

At Harvard he built the leading school of algebraic geometry in the United States. His doctoral students included Daniel Gorenstein and Maxwell Rosenlicht (1950), Shreeram Abhyankar (1956), Michael Artin and Heisuke Hironaka (1960), David Mumford (1961), and Steven Kleiman and Joseph Lipman (1965). Earlier students included Irvin Cohen and Abraham Seidenberg at Johns Hopkins, and Pierre Samuel and Robin Hartshorne, whose degrees were from Princeton. Hironaka resolved singularities in characteristic 0 in all dimensions; Mumford, Artin and Hartshorne became central figures of the Grothendieck era.

Zariski was open to new ideas that went beyond his own. He learned sheaf theory at a 1953 summer institute in Colorado, and he invited Grothendieck to Harvard, where Grothendieck’s seminars reshaped what his students learned. With Samuel he wrote Commutative Algebra (1958 and 1960), for decades the standard reference. From 1965 he turned to equisingularity: how to decide when a family of singularities has constant type, and how to stratify a variety accordingly. He was vice-president of the American Mathematical Society in 1960–61 and its president in 1969–70, and he retired from teaching in 1969.

Last years and legacy

For much of his later life Zariski suffered from tinnitus and increasing deafness, which made seminars and crowds difficult. He continued research on equisingularity into his seventies; in 1971 he also asked whether certain purely inseparable covers of the plane in characteristic \(p\), now called Zariski surfaces, must be rational when they have no holomorphic 2-forms, a question later answered negatively. From the late 1970s he was affected by Alzheimer’s disease. He died at home in Brookline, Massachusetts, on 4 July 1986, aged 87. Generations of students remembered the hospitality of the Zariskis’ home.

Honours

• Cole Prize in Algebra (1944) and election to the National Academy of Sciences (1944); American Academy of Arts and Sciences (1948); American Philosophical Society (1951); Accademia dei Lincei and the Brazilian Academy of Sciences (1958).
• AMS Colloquium Lectures (1947) and an address at the International Congress of Mathematicians (1950).
• National Medal of Science (1965).
• Wolf Prize in Mathematics (1981), shared with Lars Ahlfors, for creating the modern approach to algebraic geometry through its fusion with commutative algebra; AMS Steele Prize (1981) for cumulative influence.

Legacy

• Foundations: the Zariski topology, normality, regular local rings and formal completions are the language of algebraic geometry. Grothendieck’s schemes were built on them.
• Resolution: his programme led directly to Hironaka’s theorem; valuation-theoretic methods remain central in positive characteristic.
• Topology of curves: Zariski pairs and Alexander polynomials of plane curves are an active field.
• Students: through Mumford, Artin, Hironaka, Hartshorne, Kleiman and Abhyankar, much of late twentieth-century algebraic geometry descends from his seminar.

Connections

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

Teachers and advisors

• Guido Castelnuovo 1924

Students

• David Mumford 1961
• Abraham Seidenberg 1943
• Daniel Gorenstein 1950
• Heisuke Hironaka 1960
• Iacopo Barsotti 1945
• Irvin Cohen 1942
• Joseph Lipman 1965
• Maxwell Rosenlicht 1950
• Michael Artin 1960
• Pierre Samuel 1947
• Robin Hartshorne 1963
• Shreeram Abhyankar 1956
• Steven Kleiman 1965

Influenced by

• Bartel van der Waerden
• Emmy Noether 1935-1936
• Federigo Enriques 1920-1927
• Francesco Severi 1920-1927
• Wolfgang Krull

Influenced

• Alexander Grothendieck
• Harish-Chandra 1949-1950

Colleagues and collaborators

• André Weil 1940s
• Lars Ahlfors 1981
• Solomon Lefschetz 1927-1937

Also mentioned in this life

• Emil Artin
• Pierre Deligne
• Évariste Galois
• Felix Hausdorff
• David Hilbert
• Bernhard Riemann

Fields

• algebraic geometry 17 lives

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Oscar Zariski, MacTutor History of Mathematics.
2. D. Mumford, Oscar Zariski, 1899–1986, Biographical Memoirs of the National Academy of Sciences.
3. Oscar Zariski, Wikipedia.
4. Oscar Zariski, Mathematics Genealogy Project.
5. Oscar Zariski, Wolf Foundation laureate page.
6. C. Parikh, The Unreal Life of Oscar Zariski, Academic Press, 1991.

Mathematics

1. O. Zariski, Algebraic Surfaces, Springer Ergebnisse, 1935; second edition with appendices by S. Abhyankar, J. Lipman and D. Mumford, 1971.
2. O. Zariski, The reduction of the singularities of an algebraic surface, Ann. of Math. 40 (1939).
3. O. Zariski, Local uniformization on algebraic varieties, Ann. of Math. 41 (1940).
4. O. Zariski, Foundations of a general theory of birational correspondences, Trans. Amer. Math. Soc. 53 (1943).
5. O. Zariski, The compactness of the Riemann manifold of an abstract field of algebraic functions, Bull. Amer. Math. Soc. 50 (1944), 683–691.
6. O. Zariski, Reduction of the singularities of algebraic three dimensional varieties, Ann. of Math. 45 (1944).
7. O. Zariski, The concept of a simple point of an abstract algebraic variety, Trans. Amer. Math. Soc. 62 (1947).
8. O. Zariski, Theory and applications of holomorphic functions on algebraic varieties over arbitrary ground fields, Mem. Amer. Math. Soc. 5 (1951).
9. O. Zariski and P. Samuel, Commutative Algebra, 2 vols., Van Nostrand, 1958 and 1960.
10. H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero, Ann. of Math. 79 (1964).
11. A. Dimca, Lectures on topology of complements and fundamental groups, arXiv:math/0510049.
12. I. Shimada, A note on Zariski pairs, arXiv:alg-geom/9505007.
13. R. Hartshorne, Algebraic Geometry, Springer, 1977, chapters I–III (Zariski topology, Main Theorem, formal functions).