Federigo Enriques’s Quest to Prove the “Completeness ” Donald Babbitt and Judith Goodstein

he golden age of the Italian school of first. The first number is now called the geometric began with Antonio genus and is denoted by pg, and the second one is Luigi Gaudenzio Giuseppe Cremona and called the arithmetic genus and is denoted by pa. included among its main contributors The difference pg − pa is called the irregularity of Enrico Castelnuovo, Federigo Enriques, F and is always ≥ 0. If pg = pa we say the surface Tand . The Italian school spanned is regular. Otherwise it is said to be irregular. In nearly a century, from the unification of in modern language: pa := (aF − 1) where aF is the 1861 to Enriques’s posthumously published post- Euler characteristic of the structure sheaf or the World War II monograph on algebraic surfaces constant term of the Hilbert polynomial of F [Dieu [Enrq 49]. In the 1890s Enriques, a mathematician 74].2 who once quipped that “intuition is the aristocratic In 1885 Émile Picard initiated the study of way of discovery, rigour the plebian way” [Hodge “Picard integrals of the first kind” on an algebraic 48], and his colleague and future brother-in-law surface F, i.e., integrals whose integrands are Castelnuovo began their monumental work on the closed algebraic regular 1-forms on an algebraic birational theory of algebraic surfaces over the surface F. Subsequently he showed that there are complex numbers C.1 Severi joined them in this no Picard integrals of the first kind on smooth effort a few years later. surfaces F ⊂ P3(C) which were known to be Broadly speaking, the aspect of algebraic sur- regular. In 1901, Enriques showed that when there faces that will concern us here can be traced back are such integrals, the irregularity of F is > 0. to Rudolf Friedrich Alfred Clebsch, Arthur Cayley, These results indicate that there is an interest- and Max Noether, who in 1868–1875 introduced ing relation between the irregularity of F and the two different genera for characterizing an alge- space of Picard integrals of the first kind on F. braic surface F. The first (proposed by Clebsch) The work of these mathematicians paved the was the dimension of the space of algebraic reg- way for the Fundamental Theorem of Irregular ular (i.e., without poles or, over C, holomorphic) Surfaces—one of the triumphs of early-twentieth- 2-forms on F, a direct analog of the genus of an century algebraic geometry. The theorem states algebraic curve C, i.e., the dimension of the space that: of algebraic regular 1-forms on C. Shortly after, pg − pa = dim P(F) = b1/2 Arthur Cayley, seeking an easier way to calculate where dim P(F) is the dimension of the linear Clebsch’s genus, introduced an expression in the space of Picard integrals of the first kind on 3 degree and characteristics of the singularities of F and b1 is the first Betti number of F. The a generic projection of the surface to P3 which he Fundamental Theorem and its proof came about, hoped gave the same number. But, in fact, it turned in turn, through the combined efforts of Picard, out that the second could be negative, unlike the Enriques, Castelnuovo, Severi, and Henri Poincaré. In the modern language going back to Hodge, this 0,1 1,0 Donald Babbitt is professor of mathematics emeritus at is the assertion that h = h = b1/2. University of California, Los Angeles. His email address is In his landmark paper, “Sulla proprietà carat- [email protected]. teristica delle superficie algebriche irregolari”, Judith Goodstein is university archivist emeritus at Cal- ifornia Institute of Technology. Her email address is 2See [Mum 76], Ch. 6.C, for details on the Hilbert polyno- [email protected]. mial of F. Additional discussion of the arithmetic genus 1See Appendix A for the definitions of some of the key and its history can be found in [Kl 05], p. 243. terms of birational algebraic geometry used in this paper. 3More of this story can be found in [Kl 05].

240 Notices of the AMS Volume 58, Number 2 published in 1905 in the Rendiconti dell’Accademia of Enriques’s relentless quest that we would like delle Scienze dell’Istituto di Bologna, Enriques pro- to tell here. vided a key piece of the proof of the Fundamental Theorem [Enrq 05]. Generally referred to as the Who Was Federigo Enriques? Completeness Theorem, Enriques’s result and Expelled by Spain in 1492, many Sephardic Jews proof were of an algebro-geometric nature—i.e., found a safe haven in Tuscany, including the ances- not involving transcendental techniques such as tors of Abramo Giulio Umberto Federigo Enriques, complex analysis and topology. Francesco Severi who was born in on January 5, 1871, and offered his own algebro-geometric proof in 1905 died in on June 14, 1946. One of three also [Sev 05]. Five years later, in 1910, Poincaré children of Giacomo Enriques and Matilde Coriat, gave an independent transcendental proof of both he moved to with his the Fundamental Theorem and the Completeness family when he was seven, dis- Theorem [Poin 10]. Although both Enriques’s and playing at an early age a taste Severi’s proofs turned out to be fatally flawed, as for and numbers. Bored Severi himself showed in 1921 [Sev 21],4 neither by a homework assignment set mathematician gave up trying to find a satisfac- by his tutor that involved com- tory algebro- geometric proof of the Completeness puting the squares of numbers Theorem. From 1921 on, Enriques and Severi car- from 1 to 30, Federigo, then ried on an open feud, which spilled over into other age eleven, figured out that he areas of their professional lives. By the time the could generate the squares by English geometer Leonard Roth arrived in Rome adding successive odd num- in 1930 to study with them and Castelnuovo, rela- bers: 1, 1 + 3, 1 + 3 + 5, and tions between Enriques and Severi had completely so on. Buoyed by his discov- soured. “Either he [Severi] had just taken offence ery, he went on to calculate the or else he was in the process of giving it,” Roth squares of numbers from 1 to recalled [Roth63]. Perhaps testifying to their pug- 1,000, publishing his results in Federigo Enriques, 1930s. nacity, both ultimately joined the Fascist Party, a small pamphlet, which cost Photo courtesy of Lorenzo Severi in 1932, Enriques in 1933. him his entire savings (seven Enriques. The University of Rome geometer Fabio Con- lira). When Enriques’s daughter asked him, many forto, who served as Castelnuovo’s assistant in the years later, whether his parents had been pleased early 1930s, later organized and edited Enriques’s with his enterprise, he flashed a smile and replied, lectures on rational surfaces for publication, an ex- “They never knew about it.” The eleven-year-old ercise that gave him some insight into Enriques’s had shown even then a streak of independence he character. At a gathering of mathematicians to never lost [Enrq 47]. commemorate Enriques, held in Rome in Febru- Enriques’s mother, a Tunisian by birth, lavished ary of 1947, he noted: “The problem of conferring attention on the education of her children. The integrity upon the theorem of the characteristic se- family was well off, thanks in part to her substantial ries of a continuous complete system tormented dowry, which allowed her husband, a wealthy rug Enriques for his entire life” [Conf 47]. Unfortu- merchant, to stop working altogether after their nately, no mathematician would find such a proof marriage. Federigo and his siblings were raised during Enriques’s lifetime. For reasons possibly and remained in academic circles: Paolo Enriques, aesthetic, the Italians chafed at Poincaré’s tran- his younger brother, became a biologist; Elbina, scendental proof.5 In 1945, nine months before their older sister, married , who his death, Enriques sent a plaintive letter to the wrote and published two papers on the geometry Italian algebraic geometer Beniamino Segre, in of algebraic surfaces in 1891, the first two papers which he lamented the lack of a proof and sug- on the subject in Italy. gested that his idea of infinitely close curves of When he turned thirteen Federigo Enriques dis- higher order—an idea he had advanced in ear- covered geometry, although his mother assumed lier lectures and elaborated on in several papers he would lose interest quickly. “You know how this between 1936 and 1938—might lead to one. En- boy is,” she once wrote to her sister, “every day his riques was right (!), although he did not have the head has a new idea that lasts for about the space mathematical tools to convert his idea into an of a morning” [Enrq 47]. Despite his mother’s acceptable algebro-geometric proof. It is the story skepticism, at the Liceo di Pisa Enriques honed his taste not just for mathematics but also for 4Severi [Sev 41] claimed to have discovered a gap in both a number of other subjects—logic, epistemology, proofs while reconstructing and simplifying Poincaré’s pedagogy, and the history of science—displaying proof of the Completeness Theorem. an intellectual curiosity that he later told a col- 5See the discussion at the end of Chapter 2.4 in [Brig-Cil league could be traced back to “a philosophical 95] on this point. infection” contracted in high school [ScoDra 53].

February 2011 Notices of the AMS 241 In the summer of 1887 Enriques took and passed teachers. In 1923 Enriques became professor of the esame di licenza liceale, the final high school higher geometry at Rome, a position he main- examination. That fall, he entered the Univer- tained until the 1938 anti-Jewish racial laws sity of Pisa and the Scuola Normale Superiore, deprived him—despite his membership in the the elite teacher-training school connected to the Fascist Party, which was summarily rescinded—of university, graduating with a doctor’s degree in his university chair. The Fascist Severi, who had mathematics, with honors, in June 1891. His math- been Enriques’s assistant in Bologna, took it over; ematics teachers there included , Luigi Enriques would not regain his chair until 1944, fol- Bianchi, Vito Volterra, and Riccardo De Paolis, lowing the liberation of Rome. Castelnuovo, who professor of higher geometry, who also served as was also Jewish, had retired from teaching at Rome his thesis advisor. in 1935, but, like his other Jewish university col- Enriques’s postgraduate training began with a leagues, was barred from using the department’s year of studies and teaching at the Scuola Normale mathematics library. (1891–1892) and a second year (1892–1893) in Unlike Castelnuovo, who had switched fields Rome, where Guido Casteln- after World War I and lectured, wrote, and pub- uovo, that university’s newly lished about relativity theory and the theory of minted associate professor probability, Enriques continued to work actively of analytic and projective in algebraic geometry until the end of his life, geometry, took him under often collaborating with colleagues, students, and his wing. “Enriques was a assistants in publishing a steady stream of papers mediocre reader,” Castelnuovo (nine with Castelnuovo, four with Severi6), school later recalled, a defect that textbooks, voluminous university-level treatises, providentially led to their lecture notes, and monographs on that subject celebrated “peripatetic conver- alone. sations”; the future direction However, Enriques’s interests, as they had at of the Italian school of al- the Liceo di Pisa, ranged far and wide—from gebraic geometry evolved as the foundations of mathematics and physiological the two young mathematicians psychology to the philosophy and history of sci- crisscrossed the streets of the ence and Einstein’s theory of relativity. He founded capital [Cast 47]. In June 1893, journals (Scientia and Mathesis) and organizations After the liberation of Enriques published his first (the Italian Philosophical Society and the National Rome, Guido Castelnuovo memoir dealing with the theory Institute for the History of Science); edited, anno- played a key role in the of algebraic surfaces, Ricerche tated, and published in Italian Euclid’s Elements; rebirth of the Lincei, di geometria sulle superficie al- and contributed many articles to the Enciclopedia whose president he gebriche [Enrq 93], followed by Italiana, a massive multivolume reference set. became in 1945. a more comprehensive general As a teacher, Enriques loved nothing better Photograph courtesy of theory three years later [Enrq than to engage in his own leisurely peripatetic the Accademia Nazionale 96]. In spring 1894, after a conversations with students, in the public gardens dei Lincei. failed bid for a professorial in Bologna or under its arcades after class. When appointment at the University he moved to Rome, the labyrinthine network of of Turin, Enriques petitioned the University of paths in the Villa Borghese became his favorite Pisa for a libera docenza, which would allow him destination; he would stop there every so often, to teach there. In its report one student at that time recalls, “to trace mys- granting the request, the committee of Pisan math- terious figures on the ground, with the tip of ematicians hailed his 1893 memoir as an example his inseparable walking stick” [Camp 47]. In re- of a keen mind and an expansive talent—although marks made shortly after Enriques’s death, Fabio noting that “the precision and the rigor of the Conforto described his colleague and coauthor’s exposition leave something to be desired” [UPisa “powerful intuitive spirit” and unalterable belief 94]. Shortly afterward, Enriques passed his Abili- in “an algebraic world that exists in and of it- tazione exams with honors, another prerequisite self, independent and outside of us”—a world in for scaling Italy’s academic ladder. which “seeing” was the most important implement In 1894 Enriques joined the Bologna faculty as in a mathematician’s toolbox: Enriques “did not a temporary instructor for the projective and de- feel the need of a logical demonstration of some scriptive geometry courses, thanks in part to Vito property, because he ‘saw’; and that provided the Volterra’s intervention on his behalf. Two years assurance about the truth of the proposition in later, he took first place in a national competition question and satisfied him completely.” Conforto for the vacant chair at Bologna; that university remained his home until 1922, when he was called 6In 1907, Enriques and Severi won the prestigious Bordin to Rome to teach complementary mathematics, a Prize of the French Academy of Sciences for their work on new course designed for high school mathematics hyperelliptical surfaces.

242 Notices of the AMS Volume 58, Number 2 recalled that on one memorable occasion, when he had failed to see the truth of a statement that [Enriques] considered obvi- ous but that we had tried in vain to demonstrate logically, he stopped suddenly (we were in the course of one of the habitual walks), and instead of trying one final demon- stration he flourished his walking stick and, pointing to a puppy on a windowsill, said, “You don’t see it? For me, it is as if I said to myself, ‘I don’t see that puppy!”’ [Conf 47]. Their disagreement didn’t stop Enriques and Federigo Enriques with family and friends, Conforto from inserting that particular unproved circa 1907. Photo courtesy of Lorenzo Enriques. statement in their 1939 book, Le superficie razion- ali. Because the racial laws banned Italian Jews Publishing abroad proved equally problematic. In from publishing, Conforto’s name is the only one 1940, he had submitted an article to the Madrid on the title page. Academy of Sciences, of which he was a member, In 1941 Castelnuovo organized a clandestine but at the end of the war, because of disruptions university for Jewish students in Rome, operated in the mail service, he still didn’t know whether or in conjunction with the Fribourg Institut Tech- not it had appeared. nique Supérieure, a private school in Switzerland. In 1942 Enriques finished writing up his lectures Enriques taught a course there in the history of at Rome on the theory of algebraicsurfacesand set mathematics. Impeccably dressed (one student re- the manuscript aside until the Germans surren- called that he invariably wore gloves), soft-spoken, dered to the Allies in northern Italy in 1945. That and regal in his carriage and gestures, Enriques September, Enriques handed the eleven chapters apparently kept his students on the edge of their of Le Superficie Algebriche to the typesetters, re- seats. The Nazi occupation of Rome in fall 1943 serving the right to make any necessary additions forced the school’s closure, and both Castelnuovo or changes while the book was in press. However, and Enriques went into hiding. nine months later he was dead from a heart at- Severi, on the other hand, was well regarded tack, leaving Le Superficie Algebriche’s fate up in by the Fascist regime and had been elected to its the air. Giuseppe Pompilj and Alfredo Franchetta, Academy of Italy in 1929 (Enriques was in the Enriques’s last students, and Castelnuovo came running for the position until the last moment). to the rescue, volunteering to read the galleys, He was also the founding president of the Italian figuring that it would take only a few months of National Institute for Higher Mathematics during work. It took much longer, because of “several ob- World War II. Brieflysuspended from his university scure points”, Castelnuovo recalled—particularly post after the liberation of Rome in June 1944, in Chapter 9, which covered continuous systems Severi was subsequently absolved of any criminal of algebraic curves on irregular surfaces, includ- activity, and he resumed teaching, retiring in ing an exhaustive discussion of the status of the 1950. Beniamino Segre then moved from Bologna Completeness Theorem, the author’s famous 1905 to Rome to take Severi’s chair of geometry. proof. Indeed, more than a hint of the difficulty Although the Fascist racial laws had forced the trio faced can be gleaned from the following Enriques to step down as the longtime editor of comment, inserted as a footnote in the chapter: Periodico di Matematiche (an influential and well- thumbed magazine that published historical and The part that follows [that is, the didactic articles for secondary school teachers), status of Enriques’s theorem] has banished him from the university, and denied been left incomplete by the au- him the right to publish under his own name in thor, and the arguments developed Italy, they did not prevent him from publishing in there present many gaps; above Italy under the pseudonym “Adriano Giovannini” all, it seems opportune to bring (derived from the names of his two children, them back, because they contain Adriana and Giovanni). In 1942 he published an ideas that perhaps, suitably com- article in Periodico di Matematiche on errors in pleted, could furnish the start of a 7 mathematics and a piece in Archivio della cultura systematic account of the theory. italiana on the ideas of Galileo. After he went into hiding, however, even this publishing ceased. 7Chapter 9, p. 333.

February 2011 Notices of the AMS 243 Several pages of Chapter 9 are then devoted to A towering figure in the world of algebraic infinitely close higher-order curves, an idea that geometry from the end of the nineteenth cen- first appeared in Sulla classificazione delle super- tury until his death in 1946, Enriques made a ficie algebriche particolarmente di genere zero, a number of major contributions to the field. To volume of lectures on a par- put it another way, his quest to rescue the proof ticular category of surfaces of his Completeness Theorem represents a small that Enriques compiled and (though important) fraction of his mathematical published in collaboration with activity during his lifetime. Early in his career Luigi Campedelli in 1934. Two (1893), he constructed an (unexpected) example years later, Enriques published of an F for which pg = pa = 0 2 a paper on infinitely close but F was not birationally equivalent to P (C). The curves of higher order in existence of such a surface indicated at the very the Rendiconti dell’Accademia minimum that more birational invariants than just Nazionale dei Lincei [Enrq 36- pg and pa would be needed to classify algebraic 1] and a second paper on this surfaces in the same way that mathematicians em- themeinthepagesofthe Rendi- ployed the geometric genus to classify algebraic conti del Seminario matematico curves. (See Appendix A for the definition of a dell’Università di Roma [Enrq birational invariant.) It also sent a signal that the 36-2], which later appeared in classification itself would be more complicated. With his appointment to Three years later, in 1896, Enriques defined what abridged form in the Lincei’s Mussolini’s new Academy he called the plurigenera of an algebraic surface, journal [Enrq 37], presum- of Italy in 1929, an infinite sequence of birational invariants asso- ably so that it might reach Francesco Severi became ciated with F. Over the next eighteen years, he a wider audience. Beniamino the regime’s spokesman and Castelnuovo, often together, used the pluri- Segre challenged Enriques’s for Italian mathematics. genera to distinguish various birational classes of new proof in [Seg 38], which Photograph courtesy of algebraic surfaces. In 1914 their investigations cul- brought a quick response in the Lincei. minated in the classification of algebraic surfaces June 1938 from Enriques [Enrq 8 into four natural classes defined in terms of the 38], along with a different proof. And although behavior of their plurigenera [Enrq 14], [Cast-Enrq even Castelnuovo came close to doubting, in the 14].9 Like the Fundamental Theorem of Irregular end, the theorem’s provability, Enriques’s “life’s Surfaces, the classification of algebraic surfaces work”, as Patrick DuVal later characterized it remains one of the great accomplishments of [DuVal 49], finally appeared between covers three early-twentieth-century algebraic geometry. years after he died. After Enriques’s death, Severi recalled [Sev 57] The Completeness Theorem and the that the personality characteristics they shared Fundamental Theorem (“vivacious, pugnacious, and sometimes impul- The 1905 Completeness Theorem of Enriques, as sive”) were what had driven them apart. Their has been noted in the first part of this article, colleague, Tullio Levi-Civita, on the other hand, furnished one of the key ingredients in the proof claimed that their quarrel stemmed from com- of the Fundamental Theorem of Irregular Surfaces. peting textbooks aimed at the same scholastic Simply put, it stated that: The characteristic series market. Castelnuovo, the eldest of the trio (born of a “good” complete continuous system of curves in 1865), and by nature austere and level-headed, on a smooth algebraic surface F is complete. (Here assiduously avoided taking sides, insisting instead “good” in the old Italian terminology means not that “it was the good fortune of the Italian school superabundant, or, in modern terms, the first of algebraic geometry to have this disinterested cohomology group of divisor class should be collaboration between 1890 and 1910” [B-G 09]. In zero.10 See Appendix A for the definition of the private, Severi lashed out at his Jewish colleagues, terms used in the statement of the theorem. claiming that his work was underrated and theirs Interestingly, Severi had given an appropriate overrated. However, Beniamino Segre, Severi’s pro- condition in 1944 [Sev44].) tégé and assistant in 1927—and himself a victim of The completion of the proof of the Fundamental the 1938 racial laws—vigorously defended Severi Theorem followed quickly on the heels of that of against charges of anti-Semitism after World War the Completeness Theorem. In 1905, with q := II. dim P(F), the dimension of the Picard integrals

9 8The anti-Jewish decrees in September 1938 ended pub- See [Gray 99] for an accessible account of plurigenera lishing in Italian journals for Enriques and Segre, but and the classification of algebraic surfaces. Enriques and Severi would continue to battle it out in 10It was always clear to both Enriques and Severi that 1942 and 1943 in the pages of the Swiss Commentarii the system should be sufficiently ample in some sense. Mathematici Helvetici. Enriques mentions this several times in his 1936 article.

244 Notices of the AMS Volume 58, Number 2 of the first kind, Severi proved that pg − pa ≥ q and pg − pa = b1 − q; the latter equality was also proved shortly after by Picard. A few months later, using the Completeness Theorem in an essential way, Castelnuovo carried out the final step in the proof by establishing that pg − pa ≤ q. These two results established the Fundamental Theorem, i.e., 11 pg − pa = q = b/2. In 1921, as noted, Severi showed that both his and Enriques’s 1905 algebro-geometric proofs of this theorem were fatally flawed. Forty years later, Alexander Grothendieck [Groth 61] intro- duced the use of higher order nilpotents in the structure sheaf to define nonreduced schemes and, using them, proved a very strong existence theorem for the Picard “scheme”, the space of The first two mathematicians (l. to r.) are divisors mod linear equivalence. When he visited Beniamino Segre and Beppo Levi. The others are unidentified Italian mathematicians. Harvard, Mumford pointed out to him that, as Courtesy of Sergio Segre. an immediate corollary, this gave the long-sought algebro-geometric proof of a conjecture he had never heard of! The Remarkable Letter from Federigo The opening salvo of the last round of papers by Enriques to Beniamino Segre in 1945 Enriques and Severi on their attempted proofs of On September 11, 1945, writing from Rome, En- the Completeness Theorem took place against the riques sent a letter to Beniamino Segre in response backdrop of World War II. On November 23, 1940, to a letter Segre had sent to Enriques from Man- on the occasion of the fortieth anniversary of the chester, U.K., on August 2nd. Most of Enriques’s publication of his first original scientific paper, letter deals with his ongoing quest to vindicate his Severi submitted a ninety-three-page paper to 1905 theorem. Mussolini’s Academyof Italy (the Lincei had ceased Rome, September 11, 1945 to exist in 1939, following its formal annexation Dear Segre, by the Fascist Academy) on the general theory I thank you for your nice let- of continuous systems of curves on an algebraic ter of August 2 (I received it surface. The paper, which included a fresh attempt the other day), and I congratulate by Severi to impose his own proof of Enriques’s 1905 theorem [Sev 41], appeared in Memorie della you on your scientific activity.... R. Accademia d’Italia a year later. Now enjoying an [After a brief discussion on the enforced retirement, Enriques read it and wrote work of another Italian mathemati- a brief rejoinder pointing out an error in Severi’s cian, Beppo Levi, Enriques comes proof, which he submitted in March 1942 to the straight to the point.] Personally, Swiss journal Commentarii Mathematici Helvetici. I am especially interested in the The journal published it [Enrq 43] in 1943, but problem of a continuous system not before sending it on to Severi, who quickly on irregular surfaces which was countered with an article of his own [Sev 43]. Much discussed in my 1942 lectures on to Enriques’s annoyance, the journal’s editors surfaces, which is now in press (belying Swiss neutrality) denied his request to [Enrq 49]. This question is ex- rebut Severi’s arguments.12 In his 1943 paper, tremely delicate. I was unable at Severi once more shot down Enriques’s proof and that time to reconstruct the proof this time claimed to have proved an even more that you had indicated on the ba- general theorem. As it happens, Severi’s effort sis of the information that you had turned out to be wishful thinking, which brings us given me before my departure for to Enriques’s letter to Beniamino Segre, composed Paris [1937?]. Severi, with whom several months after German troops abandoned you have had more interaction, their hold on northern Italy and surrendered to believed he had finally succeeded the Allies. in giving a proof [Sev 41]. His exposition seems obscure to me 11Here we are using the notation introduced earlier. See and therefore dubious; I believed also [Kl 05], p. 244. (in the Commentarii Helvetici pa- 12Efforts by one of us (JG) to examine relevant documents per [Enrq 43]) to have overcome in the journal’s archives failed. the difficulty: In reality, my proof

February 2011 Notices of the AMS 245 was erroneous, but this realiza- Conclusion tion also pointed out the error Contrary to the received wisdom on the subject, in Severi’s proof. At that time, I Enriques came tantalizingly close to realizing his was not allowed to add anything quest. In the final paragraph of the last paper he to my paper in the Commentarii would publish in Italy on the Completeness The- Helvetici although Severi was al- orem, Enriques summed up his novel approach, lowed to write a note to my paper reiterated his deep-felt belief that the history of [Sev 43] in which he says that failed attempts in the history of science is instruc- he has obtained a more general tive for future researchers, and issued a gentle theorem.... But shortly afterward, rebuke to his naysayers, declaring Severi himself, who was expound- But the use of infinitely close ing that theory in his lectures at the points and curves and the cor- Institute of Higher Mathematics, realized that his proposed proof responding language, even if not was flawed due to a radical error. I rigorous, is fruitful and even suc- really wish that this thing could be ceeds in giving correct and coher- settled. I have reexamined my ear- ent results for those who can learn lier proof based on infinitely close to understand and adopt them curves of various orders and I be- as I have succeeded in doing in lieve it is substantially right, even a clear and definitive manner in if it is not rigorously complete [our other parts of my work. A skepti- italics]. cal attitude toward these ideas is In closing, Enriques asks Segre, whose August easyto havebut is not veryproduc- 2nd letter must have been written in English, a tive. Instead, those who are more language he became proficient in while living in trusting of what these concepts England during the war, to give him more details can yield, will, I am sure, discover in Italian.13 In particular, he asks Segre to limit new results in other fields. [Enrq himself to a very specific type of surface and 38] continuous system, where Enriques apparently Indeed, Enriques himself realized that he lacked expects there will be the need for infinitesimally the tools to make his infinitely close points and close curves of higher order. [See Appendix B for curves “rigorously complete”. For Enriques, the a copy of the original letter.] important thing was always to make discoveries; Later that year, Castelnuovo also urged Segre to the necessary rigorous proofs, he used to tell put his mind to the problem.He wrote many letters students, would follow in due time. It is a pity to Segre in 1945, mainly dealing with conditions that he did not live to see that he had had the in postwar Italy and urging him not to turn his right intuitive idea in the 1930s—an idea that back on his homeland, but on this occasion he eventually would lead to a rigorous proof of his tried to enlist Segre’s help in bringing closure to Completeness Theorem. Enriques’s 1905 theorem. “Given the delicacy of In an unpublished paperpreparedin2008 about the matter, a work aiming exclusively at settling Grothendieck’s work and how it affected his own the interesting question about the characteristic understanding of the algebraic geometry of En- series of a continuous system of curves would riques’s generation, the American mathematician seem appropriate,” Castelnuovo wrote [his italics]. David Mumford wrote: “Under what conditions,” he asked, Although Enriques’s 1905 paper on can one use the theorem of the the Completeness Theorem missed completeness of the characteris- the key issue, this paper [Enrq tic series of a complete contin- 38] does have the right idea. He uous system? The theorem has speaks of the exponential map in been used in fundamental investi- the Picard variety and asserts that gations on surfaces, and we would analogously higher order infinitely need to have a fully satisfying near curves can be generated from geometrical demonstration [Cast first order infinitely near ones. Un- 45]. fortunately, he possesses no tools It doesn’t appear that Segre, who turned from whatsoever for going beyond an algebraic geometry to combinatorial geometry intuitive description of why the within a few years, took the bait. method of higher order infinites- 13While organizing the Beniamino Segre papers, JG found imals should work: the theory of Enriques’s letter. There is no copy of Segre’s reply in his schemes is what he lacked. [Mum papers. 08]

246 Notices of the AMS Volume 58, Number 2 See the accompanying article by David Mum- 1/z1 : 1/z2)(= (y0 : y1 : y2)). Note that the domain −1 2 −1 ford, “Intuition and Rigor and Enriques’s Quests”, of T is P (C) −{y0y1y3 = 0} and T ((y0 : y1 : for the end of this story. y2)) = (1/y0 : 1/y1 : 1/y2). A birational invariant is an integer-valued func- Appendix A tion I from the set (category) of algebraic surfaces Birational Equivalence and Birational Invari- 14 to the integers Z such that I(F) = I(F) if F and ants F ′ are birationally equivalent. (There are similar Much of Enriques’s work dealt with the bira- definitions for algebraic curves.) tional geometry of projective algebraic surfaces In 1870 M. Noether proved that the geometric n n F ⊂ P (C), where P (C) is complex n-dimensional genus pg was a birational invariant, while Zeuthen, projective space. By we mean in 1871, proved that the arithmetic genus pa was ′ the study of birational mappings T : F “→” F of also a birational invariant. The plurigenera of projective algebraic surfaces and the correspond- Enriques are also birational invariants. ing notions of birationalequivalenceand birational invariants. Remark. Birational invariants that are also topo- Perhaps the most familiar smooth algebraicsur- logical invariants: An algebraic surface F over C is faces to nonalgebraic geometersare those given by also a topological space. As a topological space pg the locus of zeros of an irreducible homogeneous and pa are also topological invariants, i.e., if alge- ′ polynomial f on P3(C), i.e., braic surfaces F and F are homeomorphic, then ′ ′ pg = pg and pa = pa. Ff :={(z0 : z1 : z2 : z3) 3 ∈ P (C) | f (z0 : z1 : z2 : z3) = 0}, The Definition of the Terms in the Completeness Theorem where (z0 : z1 : z2 : z3) denotes the standard homogeneous coordinates in P3(C) and f is an A good algebraic curve C on F usually means it irreducible homogeneous polynomial whose dif- is sufficiently ample, that is, it is a sufficiently high multiple of a hyperplane section in some ferential never vanishes at points of Ff . It turns out that these surfaces are always regular. How- projective embedding of F. In the case of the ever, it is the smooth irregular surfaces, i.e., when Completeness Theorem, it is enough that it has no higher cohomology groups. In simplest terms, pg −pa > 0, that are most relevant for us. Irregular surfaces will thus, of necessity, be embedded in this means dim L(D) equals its self-intersection 2 Pn(C), n > 3. They will be given by the locus of number (D ) minus the genus of D plus the zeros of some set of homogeneous polynomials arithmetic genus of F plus two and that there are on Pn(C) (1) with conditions on the polynomials no regular 2-forms on F zero along D. that make sure they define an irreducible alge- A complete algebraic (sometimes called a con- braic set of dimension 2 and (2) that are locally tinuous or nonlinear) system of curves Ca, a ∈ S transverse intersections of hypersurfaces with lo- on F, where S is an algebraic variety, is maximal 16 cally independent differentials. From now on, we in the set of algebraic systems of curves on F. will suppress the reference to “complex smooth The algebraic system is good if it consists of algebraic” and refer simply to curves and surfaces. good algebraic curves. To define the notion of a birational mapping The characteristic linear system at a curve C0 ′ between surfaces F and F , we first need to define of an algebraic system, Ca, a ∈ S on F is the set the notions of a Zariski open subset in F and of divisors on C0 which are the limits of the ′ rational mapping from F to F .A Zariski open set intersections C0 ∩ Ca as a tends to 0. It is not hard in F is just the complement F − Si Wi of a finite to show that this defines a linear system on C0. set of subvarieties Wi ⊂ F.A rational mapping of The linear system on C0 is “complete” as a linear ′ F to F consists of a Zariski open subset F1 of system if it is maximal in the set of linear systems ′ 17 F and a mapping T : F1 → F , given by rational on C0. functions f of the coordinates of F. T is birational if T is injective and T −1 is rational. F and F ′ are Acknowledgments birationally equivalent if there exists a birational We thank David Mumford, who put his deep mapping between them. knowledge of the history and mathematics related ′ 2 15 2 to the Completeness Theorem at our disposal; Example. Let F = F = P (C) and F1 = P (C) − Steven Kleiman for suggesting many improve- {z0z1z3 = 0} and define T by (z0 : z1 : z2) → (1/z0 : ments, both historical and mathematical, to the 14A very nice discussion of birational algebraic geometry original text; Lorenzo Enriques for permission can be found in the paper by A. Grassi in the Bulletin of the AMS (2009), pp. 99–123. 16See [Shaf 74], Ch. I.3.5, for the definition of an algebraic 15 2 3 Here we think of P (C) embedded in P (C) by (z0 : z1 : system of curves on a surface. 17 z2) → (z0 : z1 : z2 : 0). See [Mum 76], Ch. 6A, for a discussion of linear systems.

February 2011 Notices of the AMS 247 Appendix B. Text of a letter from F. Enriques to Beniamino Segre, 11 September 1945, recounting the recent history of his efforts to prove the “Completeness Theorem”. The war had officially ended. But the censor’s stamp in the upper left-hand corner reminds us that postal authorities still opened and read letters. Courtesy of California Institute of Technology.

to publish his grandfather’s letter and informa- ed. (Agorà Edizioni, 2004). Many important tion about his family; Sergio Segre for opening scientific papers by Enriques are included in Mem- Beniamino Segre’s papers to scholars; Elisa Piccio, orie scelte di geometria, 3 vols., Bologna, Zanichelli, Michele Vallisneri, and Annalisa Capristo for dis- 1956–1966. cussions; Francesca Rosa for archival research in Rome and Pisa; and Sara Lippincott for editorial [B-G 09] D. Babbitt and J. Goodstein, Guido suggestions. Castelnuovo and Francesco Severi: Two personalities, two letters, Notices Amer. References Math. Soc. 56 (2009), 800–808. There is a vast amount of literature relating to [Brig-Cil 95] A. Brigaglia and C. Ciliberto, Italian Federigo Enriques. The starting point for under- Algebraic Geometry between the Two standing how Guido Castelnuovo and Enriques World Wars, trans. J. Duflot, Queens worked together is Riposte Armonie: Lettere di University, Kingston, Ontario, Canada Federigo Enriques a Guido Castelnuovo, edited (1995). [Camp 47] L. Campedelli, Federigo Enriques nella by U. Bottazzini, A. Conte, and P. Gario (Bol- storia, la didattica e la filosofia delle lati Boringhieri, 1996). See also Fabio Bardelli, matematiche, Per. Mat. (4) 25 (1947), “On the origins of the concept of irregularity”, 95–114. Rend. del Circolo Matematico di Palermo, sup- [Cast 45] Letter from G. Castelnuovo to B. plement 36 (1994), pp. 11–26; C. Ciliberto and Segre, December 18, 1945, Be- A. Brigaglia, “Enriques e la geometria algebrica niamino Segre Papers, Box 5, italiana”, in Dossier Federigo Enriques, ed. S. Di Institute Archives, California In- Sieno, Lettera Pristem 19–20, pp. 2–27 (March– stitute of Technology, Pasadena, June 1996); Federigo Enriques: Approssimazione CA. G. Castelnuovo F. Enriques e verità, ed. O. P. Faracovi (Belforte, 1982); Fed- [Cast.-Enrq. 14] and , Die algebraischen Flächen vom Gesicht- erigo Enriques Matematiche e Filosofia, Lettere spunkte der birationalen Transfor- Inedite, Bibliografia degli Scritti, O. P. Faracovi mationen aus, in Encyklopädie der and L. M. Scarantino, eds. (Belforte, 2001), which mathematischen Wissenschaften, B.G. includes a complete bibliography of his writ- Teubner, Leipzig (1914), III, 674–768. ings; and Intorno a Enriques, L. M. Scarantino,

248 Notices of the AMS Volume 58, Number 2 [Cast 47] , Commemorazione del socio Hilbert, 1960/61, Séminaire Bourbaki Federigo Enriques, Rend. Acc. Naz. 221. Lincei (8) 2 (1947), 2–21. [Hodge 48] W. V. D. Hodge, Federigo Enriques, [Conf 47] F. Conforto, Federigo Enriques, Hon. F. R. S. E., Royal Society of Necrologio, Rend. Mat. (5) 6 (1947), Edinburgh Year Book, 1948/49, 13–16. 226–252. [Kl 05] S. L. Kleiman, The Picard scheme. [Dieu 74] Jean Dieudonné, Cours de géométrie Fundamental algebraic geometry: algébrique I—Aperçu historique sur Grothendieck’s FGA explained, Mathe- le développement de la historique matical Surveys and Monographs, 123, géométrie algébrique, Presses Amer. Math. Soc., Providence, RI, 2005. Universitaires de France (1974). [Mum 76] D. Mumford, Algebraic Geometry I: [Du Val 49] P. Du Val, Review of [Enrq-2] in Complex Projective Varieties, Springer- Mathematical Reviews, Review number Verlag, 1976. MR0031770. [Mum 08] , My introduction to schemes and [Enrq 93] F. Enriques, Ricerche di geometria functors, unpublished paper, to appear sulle superficie algebriche, Mem. Acc. in a collection of essays concerning Torino (2) 44 (1893), 171–232. Grothendieck and his impact, edited by [Enrq 96] , Introduzione alla geometria so- Leila Schneps. pra le superficie algebriche, Mem. Soc. [Poin 10] H. Poincaré, Sur les courbes tracées It. Sci. XL (3) 10 (1896), 1–81. sur les surfaces algébriques, Ann. École [Enrq 05] , Sulla proprietà caratteristica Norm. Sup. (3) 27 (1910), 55–108. delle superficie algebriche irregolari, [Roth 63] L. Roth, Francesco Severi, J. London Rend. Acc. Sci. Bologna 9 (1904–1905), Math. Soc. 38 (1963), 282–307. 5–13. [ScoDra 53] G. Scorza Dragoni, Su alcuni para- [Enrq 14] , Sulla classificazione delle su- dossi matematici, Rend. Sem. Mat. Fis. perficie algebriche. . ., Rend. R. Acc. Milano 24 (1952–1953), 78–87. Lincei (5) 23 (1914), 206–214, 291–297. [Seg 38] B. Segre, Un teorema fondamentale [Enrq 36-1] , La proprietà caratteristica delle della geometria sulle superficie alge- superficie algebriche irregolari e le briche ed il principio di spezzamento, curve infinitamente vicine, in Rend. R. Ann. di Mat. (4) 17 (1938), 107–126. Acc. Lincei (6) 23 (1936), 459–462. [Sev 05] F. Severi, Intorno alla costruzione [Enrq 36-2] , Curve infinitamente vicine so- dei sistemi completi non lineari che pra una superficie algebrica, in Rend. appartengono ad una superficie irrego- Sem. Mat. R. univ Roma (4) 1 (1936), lare, Rend. Circ. Mat. Palermo 20 (1905), fasc. 1, 1–9. 93–96. [Enrq 37] , Curve infinitamente vicine so- [Sev 21] , Sulla teoria degli integrali sem- pra una superficie algebrica, in Rend. R. plici di 1a specie appartenenti ad una Acc. Lincei (6) 26 (1937), 193–197. superficie algebrica (7 notes), Rend. [Enrq 38] , Sulla proprietà caratteristica R. Acc. Lincei (5) 30 (1921), 163–167, delle superficie algebriche irregolari, 204–208, 231–235, 276–280, 296–301, Rend. R. Acc. Lincei (6) 27 (1938), 328–332, 365–367. 493–498. [Sev 41] , La teoria generale dei sistemi [Enrq-Conf 39] F. Enriques and F. Conforto, Le su- continui di curve sopra una superfi- perficie razionali, Zanichelli, Bologna, cie algebrica, Mem. R. Acc. d’Italia 12 1939. (1941), 337–430. [Enrq 43] F. Enriques, Sui sistemi continui di [Sev 43] , Intorno ai sistemi continui curve appartenenti ad una superficie al- di curve sopra una superficie alge- gebrica, Comment. Math. Helvetici 15 brica, Comment Helv. Math. 15 (1943), (1943), 227–237. 238–248. [Enrq 45] Letter from F. Enriques to B. Segre, Sep- [Sev 44] , Sul teorema fondamentale dei tember 11, 1945, Beniamino Segre Pa- sistemi continui di curve sopra una su- pers, Box 5, Institute Archives, Califor- perficie algebrica, Ann. di Mat. (4) 23 nia Institute of Technology, Pasadena, (1944), 149–181. CA. [Sev 57] , Parole pronunciate dal prof. [Enrq 47] A. Enriques De Benedetti, Ricordi del Francesco Severi, Rend. Mat. e Appl. (5) Babbo, Per. Mat. (4) 25 (1947), 73–80. 16 (1957), 3–8. [Enrq 49] F. Enriques, Le superficie algebriche, [Shaf 74] I. R. Shafarevich, Basic Algebraic Zanichelli, Bologna, 1949. Geometry, Springer-Verlag, 1974. [Gray 99] J. Gray, The classification of algebraic [UPisa 94] Fascicolo di Federigo Enriques, 1894, surfaces by Castelnuovo and Enriques, Archivio generale di ateneo, Università Mathematical Intelligence 21 (1999), di Pisa. 59–66. [Groth 61] A. Grothendieck, Techniques de con- struction et théorèms d’existence en géométrie algébrique IV: Les schémas de

February 2011 Notices of the AMS 249