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Images of Italian Mathematics in France pp 93–126Cite as

Birkhäuser

Picard and the Italian Mathematicians: The History of Three Prix Bordin

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Part of the book series: Trends in the History of Science ((TRENDSHISTORYSCIENCE))

Abstract

It is usually said that in the transition period between 19th and 20th centuries, French scholars (mainly Picard and Humbert ) as well as Italian scholars (mainly Castelnuovo , Enriques and Severi ) were interested in the study of algebraic surfaces, though using different methods.

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Notes

  1. 1.

    For the relations between Picard and the Italian school, see for example (Dieudonné 1974, vol. 1) and, more recently, (Houzel 2002). In the preface to the edition of the correspondence between Castelnuovo and Enriques , U. Bottazzini, A. Conte and P. Gario provide the best dealing of such relations (Bottazzini et al. 1996); in this correspondence Picard is largely the most cited mathematician, apart Segre . One of the best survey on Lefschetz’s work in algebraic geometry, the chapter by Phillip Griffths in (Griffiths et al. 1992) does not mention the influence of Scorza on the American scholar. However, a deep analysis of this influence is in (Hawkins 2013).

  2. 2.

    Letter by Enriques to Castelnuovo . The letters of the two brothers in law are published in (Bottazzini et al. 1996). In the following, we shall quote many letters from this book, without repeating the reference. The same for the letters from Picard to Castelnuovo , which are published in the same volume.

  3. 3.

    The “Quaderni” of lectures in higher geometry by Corrado Segre are available on the web site http://www.dm.unito.it/segre/quaderni.php (L. Giacardi, ed.). See also the introduction to the lectures held in 1890–1891 by A. Conte in the same web site.

  4. 4.

    For Picard’s work and mathematical definitions of the terms used by the French scholar, see chapter 10 of (Houzel 2002). It is the better reference for Picard work in algebraic geometry.

  5. 5.

    See (Picard 1885).

  6. 6.

    Indeed, Enriques never refers to them in his paper on surfaces of the same year.

  7. 7.

    He referred to (Picard 1885) and (Picard 1886).

  8. 8.

    We remark that this paper was published also in the Rendiconti del CircoloMatematico di Palermo, 9, 1895, pp. 159–166.

  9. 9.

    Letter of Enriques to Castelnuovo , 3 November 1894 (Bottazzini et al. 1996, 144).

  10. 10.

    Letter of Enriques to Castelnuovo , no dated but surely written between 3 and 11 of December 1894. (Bottazzini et al. 1996, 157–8).

  11. 11.

    Letter of Enriques to Castelnuovo , 20 January1895 (Bottazzini et al. 1996, 166).

  12. 12.

    Here I do not mean the difficulties to understand their respective kind of reasoning. Castelnuovo and Enriques were certainly able to follow a proof written in the “transcendental” language. The problem consisted in translating from one language into the other one. One of their aims was, in fact, to give a methodologically homogeneous treatment of the study of algebraic surfaces and not merely to give a set of results.

  13. 13.

    Letter of Picard to Castelnuovo , 21 January 2 1895 (Bottazzini et al. 1996, 659–660).

  14. 14.

    Letter of Picard to Castelnuovo , 22 March 22 1895 (Bottazzini et al. 1996, 660).

  15. 15.

    For the contributions of Picard on this kind of surfaces see (Houzel 2002, 217–218).

  16. 16.

    Letter of Castelnuovo to Enriques , 14 July 1895 (Bottazzini et al. 1996, 200-1).

  17. 17.

    Letter of Picard to Castelnuovo , 24 July 1895 (Bottazzini et al. 1996, 662).

  18. 18.

    See (Castelnuovo 1896). For more details on the elaboration of Castelnuovo criterion, see (Conte 1994). Here it is possible to get the flavour of this intellectual adventure, which has the pathos of a novel.

  19. 19.

    This paper had been elaborated for a very long time—from 1894 when it was requested by Nöther until March, 1896. During this time the two mathematicians hardly worked on solving the question.

  20. 20.

    (Picard 1891 – 1896) Enriques read attentively this book and was enthusiastic of it: «Sto leggendo i fascicoli usciti del 3° vol del “Traité d’Analyse” di Pic. Ci sono delle cose interessantissime! In particolare sono entusiasta della teoria delle equazioni algebrico—diff del 1° ord, e della sua integrazione pel caso che l’integrale non abbia punti critici mobili: questo problema si riduce a quello delle trasf. biraz d’una curva alg in sé.» (Letter of Enriques to Castelnuovo , 21 January 1896, Bottazzini et al. 1996, 241–2).

  21. 21.

    Letter of Enriques to Castelnuovo , 30 November 1896 (Bottazzini et al. 1996, 294).

  22. 22.

    Letter of Picard to Castelnuovo , 23 March 1897 (Bottazzini et al. 1996, 663).

  23. 23.

    Letter of Enriques to Castelnuovo , 27 March 1897 (Bottazzini et al. 1996, 326).

  24. 24.

    Letter of Enriques to Castelnuovo , without date, but written after 27 March and before 9 April 1897 (Bottazzini et al. 1996, 327).

  25. 25.

    Letter of Enriques to Castelnuovo , without date, but written after 9 April and before 22 April 1897 (Bottazzini et al. 1996, 328).

  26. 26.

    Picard denotes p1 the first number of Betti plus one.

  27. 27.

    Letter of Enriques to Castelnuovo , 14 April 1899 (Bottazzini et al. 1996, 409).

  28. 28.

    This second tome has been published in three issues, 1900, 1904 and 1906.

  29. 29.

    Letter of Enriques to Castelnuovo , March 15, 1900 (Bottazzini et al. 1996, 451).

  30. 30.

    Picard writes (p. 21): «Nous avons utilisé aussi dans notre rédaction une Mémoire de M. Segre … et une Mémoire de M. Bertini .» Picard is always very precise in his quotation and never fails to recognize the work of other scholars.

  31. 31.

    Castelnuovo and Enriques collaborated strictly—their paper appeared in the same journal and complemented each other. The problems linked with the definition and the calculation of irregularity were one of the most important topic in algebraic geometry till recent times. For a throughout discussion of it see (Bardelli 1994).

  32. 32.

    Letter of Enriques to Castelnuovo of 11 March 1902 (Bottazzini et al. 1996, 492).

  33. 33.

    Letter of Enriques to Castelnuovo of 7 March 1902 (Bottazzini et al. 1996, 490).

  34. 34.

    Letter of Enriques to Castelnuovo without date, but dated by the editors between 20 and 24 December (Bottazzini et al. 1996, 568).

  35. 35.

    This group, called the Picard group, must not be confused with the Picard group in number theory consisting in the subgroup of the Möbius group made by transformations with coefficients Gaussian integers.

  36. 36.

    It is necessary to note that the proof of the theorem of Enriques of 1904 was, many years later, considered erroneous. Luckily, in the meantime, Poincaré had found a new demonstration using transcendental methods. For the entire story I refer to the quoted paper by Bardelli.

  37. 37.

    Comptes Rendus de l’Académie des Sciences, séance du 2 décembre, 1907, pp. 981–983.

  38. 38.

    On this history, see for instance (Ciliberto and Sernesi 1991).

  39. 39.

    Letter of Enriques to Castelnuovo 2 December, 1906 (Bottazzini et al. 1996, 646–8).

  40. 40.

    Letter of De Franchis to Guccia October 21, 1907 in the Archive of the Circolo Matematico di Palermo.

  41. 41.

    Besides the “rapporteur” Georges Humbert , the members of the commission were: Jordan , Poincaré , Émile Picard , Appell , Painlevé , Maurice Levy , Darboux , Bousinnesque .

  42. 42.

    Letter of De Franchis to Guccia October 21, 1907 in the Archive of the Circolo Matematico di Palermo.

  43. 43.

    Bagnera and De Franchis had already published on 7 and 21 April a paper on the same argument in the Accademia dei Lincei, see (Bagnera and De Franchis 1907a). This time the paper was presented directly by Enriques .

  44. 44.

    See (Ciliberto and Sernesi 1991).

  45. 45.

    Comptes rendus de l’Académie des Sciences, Séance du 20 Décembre 1909, 149, pp. 1185–1188.

  46. 46.

    This name has been universally adopted and is still used.

  47. 47.

    A detailed examination of this paper of Scorza is in (Hawkins 2013).

  48. 48.

    I will not develop here the contributions by Carlo Rosati , which are really important and principally oriented towards the study of correspondences between curves. Also, many other Italian mathematicians were involved in the study of Abelian varieties. I cite here only Annibale Comessatti and Fabio Conforto. To have a deep examination of the work of Lefschetz in algebraic geometry, see the chapter by Phillip Griffiths in (Griffiths et al. 1992).

  49. 49.

    Letter of S. Lefschetz to M. De Franchis, Archivio del CircoloMatematico di Palermo.

  50. 50.

    Now I always quote from the paper (Lefschetz 1923), which is evidently a “resumé” of his lectures, written probably to clarify what Lebesgue (and some others, I guess) had not well understood during the talks.

  51. 51.

    Between 1906 and 1921, apart from the two chapters in Klein Encyclopaedia, Castelnuovo wrote only one paper on Algebraic Geometry, in a volume dedicated to Enrico D’Ovidio , his former professor. After (Castelnuovo 1921) he published only one work on algebraic geometry, after 28 years, (Castelnuovo 1949).

  52. 52.

    Segre expressed this point of view many times, particularly in his celebrated talk during the Heidelberg International Congress in 1904; see (Segre 1905).

  53. 53.

    Cited in (Parikh 2009). Zariski, coming from Russia, had studied in Rome with Enriques and Castelnuovo.

  54. 54.

    The draft of this letter by Castelnuovo is in the Archive of the Accademia dei Lincei. Now in the web, edited by P. Gario, http://operedigitali.lincei.it/Castelnuovo/Lettere_E_Quaderni/menu.htm.

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    Aldo Brigaglia

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  1. Humanités et sciences sociales, École polytechnique, Palaiseau Cedex, France

    Frédéric Brechenmacher

  2. Université d'Artoise, Lens, France

    Guillaume Jouve

  3. Laboratoire de Probabilités et Modèles Aléatoires, Université Pierre et Marie Curie , Paris Cedex 05, France

    Laurent Mazliak

  4. UFR Mathématiques, Université Lille 1 , Villeneuve d'Ascq, France

    Rossana Tazzioli

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© 2016 Springer International Publishing Switzerland

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Brigaglia, A. (2016). Picard and the Italian Mathematicians: The History of Three Prix Bordin . In: Brechenmacher, F., Jouve, G., Mazliak, L., Tazzioli, R. (eds) Images of Italian Mathematics in France . Trends in the History of Science. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-40082-2_5

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