(i) D’Orgeval’s research in the interwar period

Bernard d’Orgeval entered the École normale supérieure in Paris in 1929.Footnote ²⁴ He obtained his agrégation in 1932, and taught mathematics at the École normale supérieure in Tehran from 1932 to 1935. In 1935–1936 he did his military service at the École d’application de l’artillerie in Fontainebleau. From 1936 to 1938, thanks to a thesis grant, he worked in Rome under the direction of the mathematician Federigo Enriques. Here it is important to stress that Enriques lost his position in 1938 when the Fascist government enacted the leggi razziali (racial laws), which in particular banned Jews from holding professorships in universities, a topic I will return to later. During his stay in Italy d’Orgeval also met Oscar Chisini, as he points out in the avant-propos of his PhD thesis: “I am grateful to Mr. Chisini … for the precisions that he was kind enough to give me in person on his method of construction of multiple planes, which I used frequently”Footnote ²⁵ (d’Orgeval Reference D’Orgeval1943a, avant-propos). The “construction of multiple planes,” or ramified covers as they are usually termed today, will be the subject of the next section. In 1938, upon returning to France, he was appointed to his first position at the Lycée d’Orléans. On March 13, 1939, he spoke at the Julia Seminar, a seminar on which I will also elaborate below. As noted above, he was a POW from 1940 to 1945, and his PhD thesis in algebraic geometry, begun in Rome with Enriques, was finished in captivity in 1943.

As Michèle Pelletier (Reference Pelletier2007, 85–86) notes, to increase his chances of obtaining a position at a university, either in science or in the history of law, d’Orgeval decided to prepare for the agrégation in the history of law and undertook a thesis in Roman law under the direction of François Dumont (1900–1980), at the time a professor in Dijon, who had already supervised the thesis of d’Orgeval’s brother, Pierre d’Orgeval Dubouchet (1908-1978).Footnote ²⁶ Bernard d’Orgeval’s thesis in the history of law, submitted in 1950, was entitled L’Empereur Hadrien and was well received by the jurors.Footnote ²⁷ Following his release at the end of the war, he was finally appointed as a lecturer in Grenoble from 1946 to 1951. From 1951 to 1955 he then taught mechanics in Algiers, and from 1955 until his retirement in 1979 he was a professor at Dijon. Between 1942 and 1975 d’Orgeval published about fifty articles dealing with geometry, be it algebraic geometry (the study of surfaces or varieties of dimension 3), projective geometry over fields of non-zero characteristics, or the study of affine Desarguesian planes with a finite number of points.

To return to d’Orgeval’s activities before the Second World War, Shafer (Reference Shafer2009) notes that d’Orgeval was involved in Action française, a royalist and anti-Semitic political organization—an involvement which started at the end of the 1920s.Footnote ²⁸ Indeed, he represented Action française at the École normale supérieure, following Robert Brasillach, in the late 1920s and early 1930s. At that time Action française was popular among French lycée and university students. The then leader of the movement was Charles Maurras (1868-1952), an influential journalist and poet. Maurras called for the “restoration of the monarchy and the elimination of the influence of Judaism, freemasonry, and foreigners in France” (ibid., 22). Shafer also stresses that, when d’Orgeval “returned [from captivity] in 1945, his condemnation of the collaborators was combined with anger that the liberation also brought punishment for some, like Maurras” (ibid., 23).

The question thus arises whether d’Orgeval’s political commitments had an impact on his mathematical work. While one may not associate his royalist political views with a modern or liberal position, I claim that this in itself does not suffice to explain his mathematical position, which, if one follows Mehrtens’ categories, may be understood as being more counter-modern than modern. To recall, Mehrtens explicitly claims that such a correlation (between counter-modern political and mathematical positions) exists, though he concentrates on German mathematicians.Footnote ²⁹ In order to better understand d’Orgeval’s mathematical position, I will briefly examine his research prior to his mobilization, as this can be situated in the context of French mathematics in the interwar period. This is complemented in the next section by a survey of developments in algebraic geometry at the beginning of the twentieth century.

* * *

To better understand the mathematical context in which d’Orgeval’s work during the 1930s was situated, I turn to Juliette Leloup (Reference Leloup2009), who examines the mathematical aspect of the interwar years through the doctoral theses defended in France during this period.Footnote ³⁰ If one examines the theses submitted in Paris and classified under the heading “geometry,” then one finds, between 1925 and 1945, nine theses in algebraic geometry,Footnote ³¹ six of which were submitted between 1935 and 1945. These six doctorands were Henri Adad, Sylvain Wachs, Bernard d’Orgeval, Max Eger, Luc Gauthier, and Léonce Lesieur. Leloup (ibid., 208–209) underlines that this group of six doctorands contains another, unique subgroup. This subgroup, consisting of four students, was highly influenced by the methods of the Italian school of algebraic geometry. The four students were d’Orgeval, Eger, Gauthier, and Lesieur, all of whom employed and referenced the works of Enriques, Francesco Severi, or Corrado Segre. However, while the works of these mathematicians from the Italian school of algebraic geometry are taken as a starting point to study certain particular questions, the four PhD students do not refer to the same set of works.

Leloup (Reference Leloup2009, 208) notes that, according to Henri Cartan, d’Orgeval used the “algebraic methods of the Italian School” to prove the main theorem of his dissertation concerning the existence of certain families of algebraic surfaces of genus 1. As we will see below, this statement must be relativized, due also to the fact that Enriques and Chisini did not employ or advocate the use of such “algebraic methods.” As noted above, between 1936 and 1938, d’Orgeval studied for two years with Enriques in Rome, and in 1938 he published a paper on his research, which probably resulted from this stay (d’Orgeval Reference D’Orgeval-Dubouchet1938). This distinguishes d’Orgeval from the other doctoral students, who do not seem to have had any direct contact with Italian mathematicians. Here one can note d’Orgeval’s commitment to classical Italian algebraic geometry, which certainly shows an affinity to counter-modern mathematics, as we will see explicitly below. While d’Orgeval and his colleagues were aware of the work of Emmy Noether and van der Waerden (see: Leloup Reference Leloup2009, 224), d’Orgeval took the “classical” problems of algebraic geometry as his starting point. In doing so he was following Chisini’s methods and was also influenced by Enriques, who had a complex view of mathematics and intuition. I will return to the positioning of d’Orgeval’s mathematical work in the field of algebraic geometry in Section 4.

Another indication of d’Orgeval’s counter-modern approach can be found in the talk he gave at the Julia Seminar in 1939. As Eckes (Reference Eckes2020, 38–39) notes, the French mathematical field was strongly marked by the creation of the Julia Seminar, which between 1933 and 1939 brought together the elite of the École normale supérieure. The seminar took place at the Institut Henri Poincaré until 1938, and thereafter at the École normale supérieure. Until 1938, the Julia Seminar was dominated by the group of mathematicians Nicolas Bourbaki.Footnote ³² As is well known, Bourbaki aimed to rewrite mathematics, and their style, as Leo Corry notes, “is usually described as one of uncompromising rigor with no heuristic or didactic concessions to the reader” (Corry Reference Corry1992, 321), though Corry calls into question this narrative with respect, for example, to their concept of structure. Mehrtens, on the other hand, completely adopts this description, and presents Bourbaki as one of the representatives of modern mathematics (Mehrtens Reference Mehrtens1990, 315–326).

To return to the Julia Seminar: until 1938 the role played by the founding members of Bourbaki was undeniably central in the choice of themes treated by the seminar’s speakers (Ricotier Reference Ricotier2021, 25), and the themes during those years (algebra and topology) had an abstract orientation. However, due to the growing opposition between the founding members of Bourbaki and Gaston Julia, resulting from their divergent political and scientific positions, a drastic change of orientation occurred: the founding members of Bourbaki no longer gave talks, the audience became younger and less international, and, no less importantly, the subject of the 1938–1939 seminar (calculus of variations) was more classical than the more abstract subjects chosen in previous years (ibid., 47–48). In view of these developments, one may claim that, in its last year, the Julia Seminar distanced itself from modern mathematics.Footnote ³³ It is in this framework that d’Orgeval gave his talk at the seminar on March 13, 1939 (d’Orgeval Reference D’Orgeval1939). Indeed, d’Orgeval’s talk deals with the “geometrization” of Riemann surfaces embedded in Euclidean spaces (ibid., 1–2). However, the question discussed in the lecture concerns Riemann and Euclidean spaces in any dimension, and cannot be considered as involving an embedding in the visible three dimensional space, and nor does it turn to intuition.

If one follows Mehrtens’ modern/counter-modern categories, d’Orgeval’s talk, and after 1938, also the Julia Seminar, may be considered as distancing themselves from modern, highly abstract or algebraic approaches, a distancing which can also be seen in the works of the Italian school of algebraic geometry (more on that below). Nevertheless, one cannot characterize d’Orgeval’s work as entirely counter-modern, as we will see in the next section. But before dealing with this subject, I first want to discuss d’Orgeval’s stay at the Oflags and the conditions that enabled his mathematical research.

(ii) D’Orgeval’s research at the Oflags

As noted above, during the first months of his captivity, d’Orgeval was in Saint-Mihiel. In a report he presented in 1950 at the fourteenth International Congress of Sociology held in Rome, d’Orgeval recounts his memories and impressions of the Oflags XIII A, XXI B, and X B, noting that the Oflag XXI B in “Szubin appeared to all those who passed through there as a privileged camp” (d’Orgeval Reference d’Orgeval1950, 658). He also describes various aspects of his four years in captivity. For example, he notes that the purchasing of “books and paper disappeared after the bombings of Leipzig [on December 14, 1943],” adding that “the shortage was so great after [19]43 that matches, razor blades, toilet paper, etc. had to be distributed” (ibid., 661). Concerning the universities, he gives a general account of their formation (ibid., 664), noting, for example, that “in Sciences, the teaching of Mathematics was assured by agrégés, [and] it is only at the end that the chance of the changes of camp brought us the collaboration of [Jean] Favard, professor at the Sorbonne”Footnote ³⁴ (ibid., 665). The nature of the collaboration with Favard is not specified, however, as Favard was at Oflag XVIII in Lienz, Austria. Since d’Orgeval wrote his PhD thesis at Oflag X B, I now turn to examine this Oflag more closely.

Oflag X B was a camp for French officers located in Nienburg am Weser, Germany. It opened in May 1940.Footnote ³⁵ The creation of the “university” there can be traced back to September 1941. The “university” had three faculties: law, science, and literature, all three managed by polytechnicians. According to a report from August 1943, the law faculty had 100 students, followed by sixty at the science faculty and fifty-five at the literature faculty; the rector of the faculty of science was André Pétrus, an agrégé of mathematics (Durand Reference Durand1994, 173).Footnote ³⁶ Another report from 1943, titled “Les camps de prisonniers de guerre en Allemagne,” published by the Direction du service des prisonniers de guerre, notes not only the “important library” (Secrétariat d’État à la guerre 1943, 178), but also the POWs’ living conditions: “the [prisoners’] rooms are bright, having large windows. … In addition, they each have a private library made up of books received individually by the prisoners” (ibid., 176; see Fig. 1).

Figure 1.

A photo of one of the prisoners’ rooms in Oflag X B taken from the report “Les camps de prisonniers de guerre en Allemagne” (Secrétariat d’Etat à la guerre 1943, 177).

D’Orgeval was most probably one of the students (at least in the faculty of law, as his thesis shows).Footnote ³⁷ He is also known to have been one of the mathematics teachers (Chevaillier Reference Chevaillier2006). Moreover, d’Orgeval recounts that, during the last year of his captivity, military events and German losses

The above citation underlines that the mathematical research mainly took place between 1941 and 1944. If one recalls Leray’s development of sheaf theory at Oflag XVII A and the relative freedom and access to literature he enjoyed there, the question arises whether, with respect to the research process and how this unfolded, d’Orgeval’s research had similar traits, and how his relative isolation shaped his research. But before dealing with the mathematical works d’Orgeval wrote at Oflag X B, it is necessary to make a short detour to his law thesis, as this underlines the conditions under which his research in general was carried out. As noted, this thesis (on the Roman emperor Hadrian) was published in 1950, that is, five years after d’Orgeval’s release from Oflag X B. While d’Orgeval himself mentions only briefly the fact that large parts of the text were prepared at this Oflag, noting after the foreword that “this book was initiated in German captivity”,Footnote ³⁸ the foreword by Dumont is more informative about the conditions under which the thesis was written:

While it is well known that letters and postcards sent to and from the Oflags were read and censored, somewhat more surprising is Dumont’s claim that “the two works [theses] went hand in hand.” Surprising since, while his law thesis was not published until 1950, no other publications by d’Orgeval on this topic were made public prior to this date. As we will see, these circumstances contrast with those of d’Orgeval’s mathematical works, since several of d’Orgeval’s mathematical works were published during his captivity.

Indeed, during his captivity, d’Orgeval managed to publish three mathematical papers and to submit his thesis in mathematics. Of these four works, only two explicitly mention the fact that they were written at Oflag X B. The paper “Sur certains plans doubles non rationnels de genres p _a = p _g = 0,” presented by Lucien Godeaux, was published in 1945, but is signed at the end “Nienburg-Weser, juin-juillet 1943” (d’Orgeval [1943] Reference D’Orgeval1945, 425). D’Orgeval’s thesis was apparently brought to completion between February and April 1943, as is indicated at the end of the avant-propos. This avant-propos contains more information about his working conditions:

This opening paragraph of the avant-propos is telling, since it is not only a clear indication that the thesis was written in captivity, but also that d’Orgeval did not have access to all of the available literature. However, this does not mean that d’Orgeval worked without any access to mathematical literature. First, as may be understood from the avant-propos, it seems that Cartan and Godeaux sent d’Orgeval some of the articles and books he needed for his research. Second, in 1950 d’Orgeval states the following: “I would like to pay tribute to the memory of my teacher Professor Enriques who, although we were at war with Italy, sent me some volumes that I needed”Footnote ⁴² (d’Orgeval Reference d’Orgeval1950, 666). In view of the fact that, due to the racial laws in Italy, Enriques at that time did not hold a university position, it is remarkable that he still managed to send mathematical books to d’Orgeval. Moreover, after the bombardments of Leipzig on December 14, 1943, d’Orgeval notes that “it was no longer possible to buy books directly in Germany; from that time on, Sonderführer Lindemann, a teacher at a grammar school in Göttingen, was able to obtain books from the university library [there]. Personally, I could only obtain mathematical works; it was impossible for me to obtain epigraphic collections”Footnote ⁴³ (ibid.). Even though d’Orgeval’s remarks concern the period after December 1943 (that is, after the completion of his thesis) the last comment shows that, at least for his mathematical research, d’Orgeval was able to consult the relevant articles, though his access must have been limited. It also implicitly points toward the involvement of the Vichy regime on the one hand and the German officers at the Oflags on the other hand, which/who supported the activities taking place at the universities. But nowhere in the papers d’Orgeval wrote during his captivity does one find a hint of his personal political opinions or their influence on his research.Footnote ⁴⁴

The opening paragraph of the avant-propos is also telling since it underlines d’Orgeval’s connection to the Italian school of algebraic geometry, and more specifically to Enriques. To stress: as will be elaborated below, Enriques advocated the use of intuition in algebraic geometry, and hence, if one chooses to follow Mehrtens’ narrative, can be considered as counter-modern. D’Orgeval’s research within this mathematical framework, which will again raise the question of the degree to which d’Orgeval’s mathematical research in the Oflags can be considered as counter-modern, will be examined in more detail below.

With the exception of his law thesis, these are the only two references in d’Orgeval’s published work to his captivity or to where and how his mathematical work was written. The two other works written during this period (a one-page note published in 1942 in the Comptes rendus hebdomadaires des séances de l’Académie des sciences, presented by Élie Cartan (d’Orgeval Reference D’Orgeval1942, 431), and a longer manuscript (d’Orgeval Reference D’Orgeval1943b), which was presented by Godeaux) do not carry any indication that this research was done in Oflag X B. Here it is clear that these two papers were written not in Oflag X B but either in Oflag XXI B (where d’Orgeval was held between the end of August 1941 and the end of August 1942) or in Oflag XIII A (where he was held from the end of 1940 to the end of August 1941). As we will see below concerning the longer manuscript (d’Orgeval Reference D’Orgeval1943b), Cartan noted that it had been sent to him already in 1942. Hence, one may assume that both papers were written in Oflag XXI B.

Moreover, while the first section of a paper submitted by d’Orgeval in January 1946 and published the same year surveys d’Orgeval’s mathematical work from 1942 to 1943 (d’Orgeval Reference D’Orgeval1946, 87), no indication is made as to where this work was carried out. The avant-propos of d’Orgeval’s thesis certainly indicates that d’Orgeval was aware that captivity shaped and conditioned his work, but, in some of the other works published during these years, this remained somewhat in the background. The question of whether the decision to conceal the place of origin was made by d’Orgeval himself or by the editors of the journals in which the articles were published remains open. It is clear, however, that Cartan and Godeaux, who presented the papers, knew very well that d’Orgeval was imprisoned in an Oflag at the time. But if we return to d’Orgeval’s captivity, the question that arises is whether his relative isolation can be detected in the mathematical works themselves, that is, whether this isolation prompted unique mathematical configurations. This aspect—the inspection of the mathematical activity itself—is somewhat absent from Mehrtens’ work, and is hence introduced here to complement his account. As we will see, a consideration of the work itself under conditions of captivity and relative isolation in Germany may also lead to a reexamination of Mehrtens’ opposition between modernity and counter-modernity.

(i) A detour: Algebraic geometry during the first decades of the twentieth century

It is well known that the rise of algebraic geometry in the first two decades of the twentieth century was mostly due to the Italian school of algebraic geometry which, between the 1890s and 1930s, contributed greatly to the developing research in this field. Among the main research topics of the Italian school, particular mention should be given to the intensive research on algebraic (complex, projective) surfaces,Footnote ⁴⁵ since it became clear during these decades that algebraic surfaces exhibited some features that had no counterpart in the theory of algebraic curves. To give one example: while for algebraic curves, the rationality of a curve is equivalent to the vanishing of its genus, in 1894 Federigo Enriques discovered an algebraic surface F for which the geometric and the arithmetic genus are 0,Footnote ⁴⁶ though F was not birationally equivalent to the complex projective plane. Such a discovery implied that the task of classifying algebraic complex surfaces would be more complicated than the corresponding task for curves.Footnote ⁴⁷

This example led Enriques and Guido Castelnuovo, from the 1890s to 1914, to search for new invariants for algebraic complex surfaces in order to distinguish various birational classes of algebraic surfaces. These invariants were called the plurigenera of an algebraic surface, and in 1914 Enriques and Castelnuovo announced the classification of algebraic complex surfaces in terms of their plurigenera. One of the objects that was researched during this investigation was the branch curve of the investigated algebraic surface (a curve I will define below). This curve, as I have elaborated elsewhere (Friedman Reference Friedman2022), was couched in different epistemic configurations, which not only prompted novel research directions on such surfaces, but also led eventually to the emergence of new various mathematical configurations in which the branch curve was considered as the main object of research. In the following I will concentrate mainly on the research done around this curve, as this was one of d’Orgeval’s main objects of investigation during his captivity.

In order to better understand d’Orgeval’s research, let us first examine what a branch curve is. Taking a complex algebraic surface F of degree n given by the equation f(x,y,z) = 0, when z is of degree n, one can project it to the complex (projective) plane—for example, by the map $$(x,y,z) \mapsto (x,y)$$ . In this way, one can consider such a surface as a degree n cover of the complex plane, since for almost every point (x ₀,y ₀) on the plane there are n pre-images on the surface. Hence, the surface is considered as having n sheets, which may interchange. For example, one may consider a cover of degree 3, z ³ = R(x,y), when R is a plane curve. The branch curve is the set of all points (x,y) on the complex plane above which the surface F has less than n pre-images. A way to directly obtain the branch curve is to consider the intersection of f = 0 and df/dz = 0 and to project the resulting curve to the plane (this method was also well known during d’Orgeval’s time). Moreover, if the projection map is generic, when examining the pre-images of a local neighborhood of the branch curve, only two sheets of the surface come together and eventually interchange, interchanges which can be described by a set of permutations. One of the common ways to investigate surfaces taken by, among others, Enriques and Castelnuovo,Footnote ⁴⁸ was to look at these surfaces as covers, for example, as double covers (of the form z ² = R(x,y)), asking about their properties and how they are dependent on a given curve R. Another way was to examine the obtained set of permutations associated to the branch curve and its possible algebraic structure, a research direction initiated by Enriques.Footnote ⁴⁹

Returning to the Italian school of algebraic geometry and recalling that d’Orgeval worked with Enriques between 1936 and 1938, one should note the major role Enriques gives to intuition.Footnote ⁵⁰ Enriques emphasized the importance of intuition in both scientific and mathematical research. As early as 1894, he remarked that “we will seek to establish the postulates derived from experimental intuition of the space that appear to be the simplest for defining the object of projective geometry” (Enriques Reference Enriques1894b, 551), and that there is an “infinity of different forms of intuition”Footnote ⁵¹ (Enriques Reference Enriques1894a, 9–10). These views continue to be present in Enriques’ thought until the end of the 1930s, when he highlighted forms of intuition, including the imagination (the intuition “of what can be seen”) and the “more abstract [intuition] … which makes it possible for the geometer to see into higher dimensional space with the eyes of the mind”Footnote ⁵² (Enriques Reference Enriques1938, 173–174). Enriques’ conception of intuition was clearly influenced by Felix Klein (see, e.g., Gray Reference Gray2008, 122–123; Giacardi Reference Giacardi and Coen2012, 223–229). I stress this aspect in Enriques’ thought since it clearly positions him—if one employs Mehrtens’ categories—as a counter-modern.

Among the other mathematicians active in this school, one may name Corrado Segre and his nephew Beniamino Segre, as well as Oscar Zariski, Eugenio Bertini, and Pasquale del Pezzo. It should be noted, however, that “the Italian school was not strictly a national ‘school,’ but rather a working style and a methodology, principally based in Italy, but with representatives to be found elsewhere in the world” (Brigaglia Reference Brigaglia, Bottazzini and Delmedico2001, 189). Moreover, this school had a heterogeneous character; that is, not all of its participants can immediately be characterized as counter-modern.

This golden age of the Italian school ended toward the end of the 1920s, and some of its main mathematicians turned to research other subjects. Zariski, for example, from 1927 to 1937 focused more on topology, before moving to rewrite algebraic geometry on algebraic foundations. His work during the 1930s focused mainly on the fundamental group, looking at topological problems arising in algebraic geometry. More specifically, in several of his papers during this period he examined the fundamental group of the complement to the branch curve. This group is the set (more precisely, the group) of all loops in the (projective) plane surrounding the branch curve,Footnote ⁵³ and Zariski attempted to understand its structure and properties. However, while it is clear that Zariski employed algebraic tools to investigate his group, one cannot claim that this series of papers constituted a precursor or a modern turn in his work. Other mathematicians who delineated other research directions with respect to the branch curve were Beniamino Segre and Oscar Chisini; the latter examined this curve and its degenerations during the 1930s (Friedman Reference Friedman2022, 98–101 and 125–133). I will examine Chisini’s research in more detail below, since d’Orgeval’s research was based on it.

One of Zariski’s major publications in this period is his 1935 book Algebraic Surfaces (Zariski Reference Zariski1935), a “definitive account of the classical theory of algebraic surfaces,” which, as Parikh claims, “would convince him of the need to rewrite the entire foundations of algebraic geometry” (2009, 51). While the question of whether this book indeed contains the seeds that convinced Zariski “to rewrite the entire foundations of algebraic geometry” goes beyond the scope of this paper, it is clear that Zariski in this book criticizes the Italian school of algebraic geometry. While he delivers a systematic exposition of the Italian literature, creating an impression of a unified Italian school of algebraic geometry operating at the turn of the century, he underlines not only missing or uncertain proofs but also the (missing) epistemic value of rigor.Footnote ⁵⁴ In this sense, Zariski was pointing out the limits of the various counter-modern positions taken by the Italian school.

During the 1930s, another approach to algebraic geometry arose: that of Bartel Leendert van der Waerden, who was also known during the 1930s due to his book Moderne Algebra.Footnote ⁵⁵ Van der Waerden’s research on algebraic geometry had already begun during the second half of 1920s; as Schappacher notes, his contributions to “algebraic geometry announced a transition from the arithmetization [à la Emmy Noether, using ideal theory] to the algebraization of algebraic geometry” (Schappacher Reference Schappacher2007, 255).Footnote ⁵⁶ His ideas for an algebraic reformulation of algebraic geometry (e.g. using generic points and specializations) were presented in a series of papers, many published during the 1930s, called Zur algebraischen Geometrie, an approach which Schappacher describes as a “modest algebraization of algebraic geometry” (ibid., 270).

As noted, while the golden age of the Italian school was in decline toward the end of the 1920s, it did not disappear altogether, and its second period is characterized not only by the activity of Francesco Severi, but also by the influence, starting in the 1920s, of Mussolini’s Fascist regime on Jewish mathematicians. Zariski immigrated to the USA in 1927 due to the increasingly alarming situation in Fascist Italy; Enriques, as is well known, was forced to resign in 1938; and B. Segre was expelled from Bologna university. The extent of the influence of the racial laws on Jewish mathematicians has been analyzed thoroughly in several publications (Finzi Reference Finzi and Joshua2005; Capristo Reference Capristo and Joshua2005).Footnote ⁵⁷ It is not without relation to these developments that the most original ideas of the Italian school during this period can mostly be attributed to contributions from “minor” authors. These mathematicians were isolated from other mathematical communities partly through their own choice (i.e. not to follow the discussions and developments in modern algebra), and partly due to the Fascist regime, which isolated the scientific and hence mathematical community from having contact with other communities outside of Italy. “Minor” refers to an epistemic configuration that deals with objects and techniques that were considered—either at that time or later—as representing marginalized or even outdated research programs.Footnote ⁵⁸ Such is the work, between the 1930s and the 1950s, of Chisini and his students, who chose not to take into consideration other, more algebraic research directions when developing their own research, research which did not, or could not, take into account the rise of modern algebraic geometry.

As we will see when examining d’Orgeval’s work between 1938 and 1945, d’Orgeval was influenced both by Chisini’s research topics and by his methods of inquiry. While I will elaborate on the mathematical configuration of Chisini himself below, here I would like to discuss his views of mathematical practices. Carlo Felice Manara, one of Chisini’s students, noted Chisini’s approach to mathematics, which not only emphasized the “vividness of the imagination, understood as spatial imagination,” but also expressed “mistrust in front of every formal reasoning and every algorithmic acrobatics” (Manara Reference Manara1987, 15). Chisini researched a variety of subjects, including complex algebraic surfaces as coverings, their branch curves, and singular curves in general. In previous works (Friedman Reference Friedman2019), I have described Chisini’s research on new modes of presentation of plane singular curves with the help of braids, though not with Artin’s algebraic formulation of braids, which was published in 1926. Here, however, it is essential to emphasize that Chisini already underlined, in an article from 1933 that working with curves and their associated braids becomes more manageable when one uses “material models” with “material threads with notable thickness” (Chisini Reference Chisini1933, 1146–1147). Both Piera Manara and Maria Dedò, the daughters of Chisini’s students,Footnote ⁵⁹ recall Chisini’s string models being of different colors used to model curves (Friedman Reference Friedman2022, 128). C. F. Manara notes that Chisini’s “distrust [in algebra] led him to want to build tangible and material models” (Manara Reference Manara1987, 23). These material models of braids were used not only to illustrate complex curves but also to prove certain properties concerning their singularities, as was later developed by Modesto Dedò. Chisini’s skepticism also led him to ignore Artin’s research (and subsequent research) on the braid group, and may have prompted a disconnect from other research directions that were on the rise during this period, such as commutative algebra.

Here it might be worth pausing to reexamine this episode with Mehrtens’ categories. Clearly, the rejection of and objection to the structural turn in algebra and the algebraic rewriting in algebraic geometry characterizes Chisini’s (and his students’) work as counter-modern in a way which is more radical than Enriques’ counter-modern approach. In a paper on material mathematical models published after Moderne—Sprache—Mathematik, Mehrtens notes that, before becoming merely pedagogical objects, these models were, at least for a certain period (i.e. during the last decades of the nineteenth century), “boundary objects” (Mehrtens Reference Mehrtens2004, 301), that is, objects to be found between modern and counter-modern mathematics. More precisely, Mehrtens notes that “mathematical models are creations of human hands and minds,” and hence may also be considered as modern mathematics, “but with [Felix] Klein they stand for Anschauung and thus for a specific brand of counter-modernism” (ibid., 293). Moreover, as Epple (Reference Epple, Otte and Panza1997) points out, Artin himself combined modern and counter-modern elements in his 1926 paper. While Artin’s work was later reconceptualized as “too modern” by Chisini (as Chisini employed his own material models and braid diagrams), the work of Dedò,Footnote ⁶⁰ who in 1950, following Chisini, eventually tried to develop “an algebra” (Dedò Reference Dedò1950, 228) of braids and their factorizations, shows that such a strict opposition between the two categories is hardly possible or helpful. We will see in the following that d’Orgeval’s work, done in a relative isolation, may also lead to a reexamination of this opposition.

On a more international and less “minor” scale, last but certainly not least in the rapidly changing landscape of algebraic geometry during the 1930s and the 1940s, one must also mention, along with Zariski’s work, André Weil’s own rewriting of algebraic geometry, reaching one of its highlights with the publication of his Foundations of Algebraic Geometry in 1946, a work begun in the 1940s. Both of these rewriting projects can be described as modern.Footnote ⁶¹ These developments will not be discussed here, however, since during his captivity in the Oflags d’Orgeval would not have had access to or knowledge of these rewritings.

As was already noted, and as I will examine more thoroughly below, d’Orgeval’s work before and during his captivity in the Oflags is situated in the framework of the Italian school of algebraic geometry. Leloup (Reference Leloup2009, 224) notes that French mathematicians (and mathematics doctorands) knew the works of van der Waerden and Emmy Noether, and in the four theses on algebraic geometry noted above (by d’Orgeval, Eger, Gauthier, and Lesieur), one finds several references to German mathematicians. While d’Orgeval most probably knew van der Waerden’s works (as van der Waerden is cited by his colleagues at that time), none of the algebraic rewriting projects (by van der Waerden, Zariski, or Weil) are either used or mentioned by him. On the other hand, all of these four theses refer heavily to the works of Italian geometricians. It is against this background that I want to analyze d’Orgeval’s work between 1938 and 1945. As we will see, his work is characterized not by a critique of lack of rigor or of Chisini’s distrust in algebra, but by a certain distance from an extreme version of opposition to “modernity” (à la Mehrtens), pointing toward the limits of Chisini’s methods. As will be highlighted below, however, reaching these limits does not necessarily mean that a “natural consequence” for the research of algebraic geometry would be a “profound remodeling of algebra” (Schappacher, Reference Schappacher2007, 247).

(ii) On d’Orgeval’s mathematical configuration: Algebraic geometry at the Oflags

D’Orgeval’s investigation of algebraic surfaces, considered as ramified covers of the complex (projective) plane, most likely began during his stay in Italy between 1936 and 1938, though his talk at the Julia Seminar on March 13, 1939 was on a completely different topic. His investigation of ramified covers arose out of his investigations of infinitesimal deformations of a degenerated curve into another curve, the latter being the branch curve of the investigated surface. The aim of his research was to understand the properties of algebraic surfaces (e.g., those whose genera are all equal to 1, the subject of his PhD thesis), which is why d’Orgeval’s research focused on a variety of ways to examine this curve, including via deformations.

While the formulation used above (“branch curve”) is a modern one, the terms d’Orgeval employs are similar: a ramified surface is called a “plan multiple” or a “plan multiple d’ordre n” (d’Orgeval Reference D’Orgeval1943a, 73), meaning that it is an algebraic surface, being a cover of order n. A branch curve is termed by d’Orgeval a “ligne de diramation” or a “courbe de diramation” (ibid.). These terms were also employed by Enriques and Chisini. Indeed, in a note published in 1938, d’Orgeval stresses that he is following Chisini’s method concerning this kind of construction using a degenerated curve: “Mr. O. Chisini has shown that it is possible to obtain a curve , which is a branch curve of an n-fold plane [cover], from a sequence of n – 1 curves linked in a suitable way”Footnote ⁶² (d’Orgeval Reference D’Orgeval1938, 1866). Moreover, d’Orgeval was not only citing Chisini’s writings; he also knew Chisini personally, as was noted above.

Here, in order to understand what Chisini’s method is and how d’Orgeval made use of it, but also modified it during his time in the Oflags, another detour is needed. This method consists of taking a curve $\overline \varphi $ with n – 1 irreducible components ${C_1}, \ldots ,\;{C_n}$ , denoted as $\overline \varphi = {C_1} \cdot {C_2} \cdot \cdots \; \cdot \;{C_n}$ . Chisini looked at the square of this curve $\overline \varphi ^2$ and claimed that with “infinitesimal variations” of $\overline \varphi ^2$ , which satisfy certain conditions, one can obtain another (irreducible) curve , which is a branch curve of an n-degree cover (Chisini Reference Chisini1934). For example, if is a curve having two components, C ₁ and C ₂ (denoted as $ =\hskip -1pt {C_1} \cdot {C_2}$ ) such that C ₁ = {f ₁(x,y) = 0} and C ₂ = {f ₂(x,y) = 0}, then $\varphi ^2$ is defined by the equation (f ₁(x,y) f ₂(x,y))² = 0. An example of an “infinitesimal variation” may be, for example, the passage of the curve xy = 0 to curve xy = ε when ε > 0 (in d’Orgeval’s research one performs an “infinitesimal variation” of $\varphi ^2$ ). Note that the curve (xy = ε) is smooth, while the limit curve (xy = 0) is singular and has a node.

One of the initial conditions of this method is that, for each component C _i of the initial curve $\overline \varphi $ , one associates the following permutation $\left( {i,\;i + 1} \right)\;$ in the symmetric group of n letters, a permutation that eventually describes which sheets of the n-degree cover are branched along the obtained branch curve.Footnote ⁶³ D’Orgeval wished to introduce certain “modifications of this construction” (d’Orgeval Reference D’Orgeval1938, 1866) and to show that other permutations may be associated (not necessarily of the form $\left( {i,\;i + 1} \right)$ ), while still employing Chisini’s method, in order to obtain other surfaces.Footnote ⁶⁴ And indeed, in 1938, he presented a diagram of curves (see Fig. 2), a diagram which does not appear in any of Chisini’s works on the subject. This diagram describes, for each curve Γ, C, or D (being a cubic curve, a conic, or a line respectively), which permutation is associated to it. To be precise, for every such diagram, the following curve is associated: $\overline \varphi =\Gamma \cdot C \cdot D \cdot \cdots \cdot C \cdot D$ (when all of the curves C or D are different from each other); afterward, the curve $\overline\varphi ^2$ is deformed, obtaining another curve denoted by , which is a branch curve of the desired surface. What is new in d’Orgeval’s 1938 paper is not only the diagram, but also how the permutations are associated to the curves: while before (in Chisini’s configuration), for each component C _i of the initial curve $\overline\varphi $ , one associates the permutation $\left( {i,\;i + 1} \right)$ , now the association of permutation to the curves Γ, C, or D is different, since, for several curves, a permutation of the form $\left( {i,\;i + 2} \right)$ is associated.

Figure 2.

One of the diagrams in d’Orgeval Reference D’Orgeval-Dubouchet1938, 1867. Note that a line connects both curves if the associated permutations share one value.

It is exactly this line of investigation that was developed further by d’Orgeval during his captivity, when he introduced several unique methods and terms. As noted above, in 1942 d’Orgeval published a one-page note that was presented by Élie Cartan. Cartan remarks in a footnote that the “present note is a summary, which attempts to be as faithful as possible, of a more developed paper transmitted by the author”Footnote ⁶⁵ (d’Orgeval Reference D’Orgeval1942, 431). Nevertheless, it is not clear whether the initiative to publish this summary came from Cartan or from d’Orgeval himself. The paper Cartan refers to is the more elaborate paper (d’Orgeval Reference D’Orgeval1943b), which was published in the following year and consists of eighteen pages. Indeed, the one-page note (d’Orgeval Reference D’Orgeval1942) is a highly condensed summary of this long paper, and as a result it may seem somewhat imprecise. It is in this publication, however, that a new term is coined for the first time to denote the surface resulting from the special arrangement of the curves presented above: “a multiple plane of entangled sheets” (“plan multiple à feuillets enchevêtrés”) (d’Orgeval Reference D’Orgeval1942, 341), meaning that the sheets of the examined surface are entangled because the associated permutations are not successive (i.e. some of them are not of the form $\left( {i,\;i + 1} \right)$ ). A diagram of three lines, all denoted by C ₁, is drawn as well, together with the associated permutations, to depict what is meant by “entangled sheets” (see Fig. 3 (left)).Footnote ⁶⁶

Figure 3.

In diagram (a) (left), which is found in d’Orgeval Reference D’Orgeval1942, 341, each of the C ₁’s is a different line. For this specific construction, d’Orgeval notes that it corresponds to the Veronese surface of degree 4. In d’Orgeval Reference D’Orgeval1943b, 225, both diagram (a) and diagram (b) are presented. Diagram (b) (right) presents the arrangement of curves associated to a ruled surface of degree 4. As before, each of the C ₁’s is a different line. © Graphic by M.F.

The paper (d’Orgeval Reference D’Orgeval1943b) is indeed a more coherent presentation of d’Orgeval’s construction, and also contains a more detailed and precise explanation of Chisini’s method (ibid., 218–222) and how it can be modified. Here another new term appears: while the term “plan multiple à feuillets enchevêtrés” (ibid., 225) is again mentioned to denote the surface resulting from the more complicated arrangement of curves, the cover which results from Chisini’s arrangement is termed a cover “of successive sheets” (“à feuillets successifs”) (ibid., 227). Both arrangements (“feuillets enchevêtrés” and “feuillets successifs”) are depicted, as shown in figure 3(left) and 3(right) respectively. Moreover, d’Orgeval stresses not only that Chisini’s method can be modified, but that this method may cause difficulties for such research on surfaces (ibid., 653).

D’Orgeval’s thesis, titled “Les surfaces algébriques dont tous les genres sont 1,”Footnote ⁶⁷ which he wrote in Oflag X B, deals, among other subjects, with the construction of surfaces according to the above method (d’Orgeval Reference D’Orgeval1943a, 73-94). While summarizing the methods described above, and again using the terms “feuillets enchevêtrés” and “feuillets successifs” (ibid., 75), d’Orgeval aims to show how, for the investigated surfaces (surfaces whose genera are all 1), these curve diagrams can be manipulated. This aspect is certainly new, and its introduction stands in a certain contrast to Chisini’s method, as I will underline below.

As d’Orgeval notes, if one takes a surface whose genera are all equal to 1, and on it a pencil of hyperplane sections, being curves, then these curves are of genus π and of degree 2π – 2 (ibid., 76). D’Orgeval also shows that the result of his thesis is that “for every given value of π, there exists one family” of such surfaces (ibid., 4). What d’Orgeval concludes from this result is that, for a given value of π, if one has two different diagrams of curves, when from both one can construct ramified surfaces (both with all of their genera being 1 and with the same given value of π), then these diagrams are equivalent, in the sense that one can operate on them and still obtain a surface in this family. For example, for π = 4, d’Orgeval notes that “we can obtain two types of representation,” which are shown in figure 4 (ibid., 82). But since there is only one such surface in the same family,Footnote ⁶⁸ “we hence conclude from that that the transport of the group C ₂ – C ₁ [being two curves found in the diagram] can be done on the multiple plane [the cover] without changing the nature of the surface” (ibid.). The expression “the transport of the group C ₂ – C ₁” refers explicitly to an action on the diagram itself: moving the set of two curves C ₂ – C ₁ from one place in the diagram to another,Footnote ⁶⁹ hence obtaining another diagram (see Fig. 4). Therefore, it is a diagram which can be manipulated. This action may, on the one hand, make the resulting diagram easier to understand. On the other hand, according to d’Orgeval, it does not change the characteristics of the surface obtained. Similar actions on diagrams are performed on surfaces with π = 5 or with π = 6, as can be seen in figure 5.

Figure 4.

Two equivalent diagrams for surfaces with π = 4 (d’Orgeval Reference D’Orgeval1943a, 82). Permission to reprint granted by the publishing house Dunod and authorized by the heirs of Bernard d’Orgeval.

Figue 5.

Several equivalent diagrams for surfaces with π = 6. Here d’Orgeval notes: “in the same way, one can show the identity of the represented surfaces” (d’Orgeval Reference D’Orgeval1943a, 83). Permission to reprint granted by the publishing house Dunod and authorized by the heirs of Bernard d’Orgeval.

* * *

If we return to Mehrtens’ differentiation between modern and counter-modern approaches to mathematics, it may be somewhat difficult to apply this classification to d’Orgeval’s research during his captivity. On the one hand, we have seen that d’Orgeval attempted to develop Chisini’s method. As was noted above, during his own research on branch curves, Chisini used material models of strings to prove certain results, a method which can be termed intuitive as well as material, and hence can certainly be classified as counter-modern, since Chisini himself often expressed his mistrust of every form of “formal reasoning” (Manara Reference Manara1987, 15). D’Orgeval did not employ those methods and hence can be considered to be attempting to introduce more modern approaches and tools to Chisini’s configuration. His use of manipulable diagrams shows this distancing from Chisini’s strict, extreme counter-modern approach.Footnote ⁷⁰

While the diagrams may also be considered as a visual, even intuitive method to represent surfaces, as if their manipulation happens in the mind of the mathematician (hence associating to her certain cognitive abilities), d’Orgeval’s investigation relied on the manipulation of this symbolic means of representation, a manipulation which takes place on the paper (i.e. as a formal procedure). I claim that this manipulation is not similar to manipulation of diagrams of knots (e.g. in our imagination) or of braids (e.g. with materials models). This is because the diagrams do not represent an object similar to themselves (as knot diagrams represent knots or Euclidean diagrams represent rectangles in certain arrangements).Footnote ⁷¹

What is left unexplained by d’Orgeval is the equivalent of this manipulative (and at the same time formal) procedure on the level of the surfaces (e.g. which birational mapping transforms the first surface with π = 4 into the second one), which are the object of investigation. Instead, with his manipulation of diagrams, d’Orgeval turns to a manipulation of symbols written on paper, hence introducing, following Mehrtens’ terms, a more modern approach to this investigation. While one may indeed claim that there is a diagrammatic reasoning in action here, it is not a purely counter-modern one, since it does not solely turn to intuition or imagination, but rather takes place as a symbolic process. This certainly echoes Mehrtens’ understanding of mathematical modernity; Mehrtens claims on numerous occasions that “modernity orients itself towards the signifiers, which it interprets as signs on paper that can be treated empirically,” (i.e. manipulated on the paper), without turning to or postulating a unified subject with “Ur-Intuition or the gift of Anschauung” (Mehrtens Reference Mehrtens1990, 414).

While the above discussion shows that d’Orgeval’s work may lead to a reexamination of Mehrtens’s modern/counter-modern distinction, I claim that d’Orgeval’s unique research configuration, developed at the Oflags, was ephemeral, an aspect which was not discussed by Mehrtens. Indeed, this unique configuration was formed during captivity; moreover, it was short-lived. This type of diagrammatical reasoning might have proved to be a novel method to study algebraic surfaces, but d’Orgeval abandoned it in 1946. To recall, d’Orgeval himself notes that, during the last year of his captivity, various aspects of the captives’ intellectual lives disappeared, forcing them to discontinue the research and eventually to stop with the activities of the university. Moreover, while similar diagrams appear in his 1946 paper, which deals with similar topics (namely, with how to represent covers with such diagrams, and the conclusions which may be drawn), such operations on the diagrams are no longer mentioned or used. In addition, the terms “feuillets enchevêtrés” and “feuillets successifs” are not even mentioned once, and in fact, after 1943, are no longer employed either by d’Orgeval or by any other mathematician.Footnote ⁷²

One of the reasons for d’Orgeval’s abandonment of his own methods and terms may lie in his conclusion concerning Chisini’s method: “The method of Mr. Chisini, in spite of the few generalizations that I added to it, still allows the representation of many surfaces to escape, even for rather small values of their [associated] invariants”Footnote ⁷³ (d’Orgeval Reference D’Orgeval1946, 101). That is, d’Orgeval distances himself from Chisini’s “counter-modern” position. Nevertheless, he presents another kind of counter-modern approach, one that acknowledges the limits of radical counter-modern approaches (such as Chisini’s) within mathematical configurations. But does this indicate an acknowledgement of the limits of working in isolation?

On the one hand, d’Orgeval’s relative isolation in the Oflags and his dependence on the cooperation of the German officers there shaped, but also narrowed, his research. On the other, it also led to the formation of another kind of counter-modern research, at least in the way d’Orgeval handled diagrams and diagrammatic reasoning. It shows, as Epple emphasized when investigating Artin’s research on braid theory, not only that there are various shades of counter-modernity (and modernity), but also that ephemeral configurations, such as those formed in the Oflags, may prompt unique combinations and versions of counter-modern and modern practices.