# Oscar Zariski

### Quick Info

Kobrin, Russian Empire (now Belarus)

Brookline, Massachusetts, USA

**Zariski**'s work was on foundations of algebraic geometry using algebraic methods. He worked on the theory of normal varieties, local uniformisation and the reduction of singularities of algebraic varieties.

### Biography

**Oscar Zariski**'s father was Bezalel Zaritsky and his mother Hannah Zaritsky. Oscar, born into this Jewish family, was named Ascher Zaritsky by his parents. We will comment below on how his name came to be changed to the now familiar version of Oscar Zariski.

Oscar's mother was the one who ensured that her young son had a good education. A tutor was provided for Oscar from the time he was seven years old and Oscar, under the guidance of the tutor, showed remarkable aptitude for the Russian language and for arithmetic. When the fighting associated with World War I reached Belarus, Oscar's family fled to Chernigov in the Ukraine. It was the first of several moves forced on him by political problems.

Unable to enter the faculty of mathematics at the University of Kiev as all the places were full, he chose philosophy instead. He was a student of philosophy at Kiev from 1918 to 1920. However he was able to pursue his mathematical interests and studied algebra and number theory in addition to philosophy. However Zariski had carried out his studies through a period of turmoil in Kiev.

In January 1918 Ukraine had become an independent state with Kiev as its capital. In the following month minor uprisings by workers in Kiev were suppressed but Red Army troops entered Kiev to give support to the workers. Kiev was then occupied by the Germans, but with the end of the war in November 1918, an independent Ukraine was declared again in Kiev. In November 1919 Kiev was briefly taken by the White armies, soon after to be replaced by the Red Army. There then followed the Russian-Polish War and, in May 1920, the Polish army captured Kiev but were forced out in a counterattack. Life for Zariski was just too difficult in this city so devastated by war, so he decided to go to Italy to continue his studies.

In Rome Zariski came under the influence of the great algebraic geometers Castelnuovo, Enriques and Severi. He obtained a doctorate from Rome in 1924 for a doctoral thesis on a topic related to Galois theory which was proposed to him by Castelnuovo. Zariski married in 1924; having met his future wife Yole Cagli in Rome they returned to Zariski's home town of Kobin to marry. Returning to Rome he remained there as a fellow of the International Education Board until 1927.

It was while Zariski was in Rome that Enriques suggested that Ascher Zaritsky, as he was then called, change his name to the Italian sounding Oscar Zariski. This was the name which he used on his first publication which was a joint paper with Enriques. Zariski wrote in [2] of how his mathematical interests differed from those of his supervisors Castelnuovo and Enriques:-

However, even during my Rome period, my algebraic tendencies were showing and were clearly perceived by Castelnuovo who once told me: "You are here with us but are not one of us." This was said not in reproach but good naturedly, for Castelnuovo himself told me time and time again that the methods of the Italian geometric school had done all they could do, had reached a dead end, and were inadequate for further progress in the field of algebraic geometry.Zariski had gone to Italy to escape the problems in Belarus and the Ukraine. However, the political situation in Italy began to deteriorate rapidly. In October 1922 Mussolini organized the Fascist "March on Rome" and he was asked to form a government. For 18 months he ran the country in reasonably democratic way but, during the years 1925 to 1927, he removed the right of free speech, and removed opposition parties and trade unions. The Fascist hatred of Jews made life for Zariski, because of his Jewish background, particularly difficult.

Helped by Lefschetz, he escaped from the political problems of Italy in 1927 and went to the United States. There he taught at Johns Hopkins University, being a Johnston Scholar until 1929 when he joined the Faculty. He became a full professor at Johns Hopkins in 1937.

Castelnuovo and Severi had encouraged Zariski to view Lefschetz's topological methods as being the road ahead for algebraic geometry, so between 1927 and 1937 Zariski frequently visited Lefschetz at Princeton. Zariski wrote [2]:-

I owe a great deal to [Lefschetz] for his inspiring guidance and encouragement.During this period Zariski wrote

*Algebraic Surfaces*which was published in 1935. He explains in [2] how writing this monograph changed the direction of his work:-

At that time (1935) modern algebra had already come to life (through the work of Emmy Noether and the important treatise of B L van der Waerden), but while it was being applied to some aspects of the foundations of algebraic geometry by van der Waerden ... the deeper aspects of birational algebraic geometry ... were largely, or even entirely, virgin territory as far as algebraic exploration was concerned. In [Algebraic Surfaces] I tried my best to present the underlying ideas of the ingenious geometric methods and proofs with which the Italian geometers were handling these deeper aspects of the whole theory of surfaces ... I began to feel distinctly unhappy about the rigour of the original proofs (without losing in the least my admiration for the imaginative geometric spirit that permeated these proofs); I became convinced that the whole structure must be done over again by purely algebraic methods.At Johns Hopkins University between 1939 and 1940 Zariski carried out his project of applying modern algebra to the foundations of algebraic geometry. He worked on the theory of normal varieties, local uniformisation and the reduction of singularities of algebraic varieties.

An important year for Zariski was 1945 which he spent in São Paolo. There he gave a lecture course three days each week which was attended by André Weil and nobody else. Both Zariski and Weil learnt much in discussions, often arguments, about the material that Zariski was presenting. After spending the year 1946-47 at the University of Illinois, Zariski was appointed to a chair at Harvard where he was to remain until he retired in 1969. From the late 1970s he suffered from Alzheimer's disease and his last few years were difficult ones as his health failed.

In 1981 Zariski was awarded the Steele Prize by the American Mathematical Society for the cumulative influence of his total mathematical research. The citation for the prize summarised Zariski's contributions to mathematics throughout his life [2]:-

After beginning his work in Italy in 1924 very much in the style of "Italian algebraic geometry," Zariski realised that the whole subject needed proper foundations. Thus in the period 1927 to 1937 he turned first to topological questions and then in 1937 he began to lay the commutative algebraic foundations of his subject. His topological work concentrated mainly on the fundamental group; many of the ideas he pioneered were innovations in topology as well as algebraic geometry and have developed independently in the two fields since then.Zariski's most famous book is

In 1937 Zariski completely reoriented his research and began to introduce ideas from abstract algebra into algebraic geometry. Indeed, together with B L van der Waerden and André Weil, he completely reworked the foundations of the subject without the use of topological or analytic methods. His use of the notions of integral independence, valuation rings, and regular local rings, in algebraic geometry proved particularly fruitful and led him to such high points as the resolution of singularities for threefolds in characteristic 0 in 1944, the clarification of the notion of simple point in 1947, and the theory of holomorphic functions on algebraic varieties over arbitrary ground fields. The theory of equisingularity and saturation begun by Zariski in 1965 has also been of great influence and importance.

All of Zariski's work has served as a basis for the present flowering of algebraic geometry and the current school uses his work and ideas in the modern development of the subject.

*Commutative Algebra*, a two volume work written jointly with P Samuel. The first volume appeared in 1958, the second in 1960.

The American Mathematical Society played a large role in Zariski's life and he contributed greatly to the Society over many years. He was vice president of the Society between 1960 and 1961 and president of the Society from 1969 to 1970.

Zariski played an important role in mathematical publishing after his appointment as a full professor. He was an editor of the

*American Mathematical Journal*from 1937 to 1941, served as a member of the editorial committee of the

*Transactions of the American Mathematical Society*from 1941 to 1947, and also served on the editorial boards of the

*Annals of Mathematics*and the

*American Journal of Mathematics*.

After going to the United States in 1927 Zariski spent considerable periods lecturing at other universities both in the United States and in other countries. We have already mentioned that he was a visiting professor at São Paulo in 1945 and a visiting professor at the University of Illinois in 1946-47. Before that, in 1936, he had lectured at the University of Moscow. Later, he lectured at Kyoto (1956), the Institut des Hautes Études Scientifiques (1961 and again 1967), and the University of Cambridge (1972).

He was awarded many honours for his work in addition to the Steele Prize described above. He was awarded the Cole Prize in Algebra from the American Mathematical Society in 1944 for four papers on algebraic varieties, two published in the

*American Journal of Mathematics*in 1939 and 1940, and the other two in the

*Annals of Mathematics*also one in 1939 and the second in 1940. He was awarded the National Medal of Science in 1965.

Many academies and societies have honoured him by electing him to membership, including the U.S. National Academy of Sciences (1944), the American Academy of Arts and Sciences (1948), the American Philosophical Society (1951), the Brazilian Academy of Sciences (1958), and the Accademia dei Lincei (1958).

### References (show)

- C Parikh,
*The unreal life of Oscar Zariski*(Boston, 1991). - 1981 Steele Prizes,
*Notices Amer. Math. Soc.***28**(1981), 504-507. - P Blass, The influence of Oscar Zariski on modern algebraic geometry,
*Austral. Math. Soc. Gaz.***16**(6) (1989), 157-168. - D Mumford, Oscar Zariski: 1899-1986,
*Notices Amer. Math. Soc.***33**(6) (1986), 891-894. *Oscar Zariski : Collected Papers*(Cambridge, Mass., 1987).

### Additional Resources (show)

Other websites about Oscar Zariski:

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Written by J J O'Connor and E F Robertson

Last Update February 2000

Last Update February 2000