Guido Castelnuovo: the lectures of Advanced Mathematics (1927–1928).

*(Italian. English summary)*Zbl 1390.01014The archives of Rome University preserve Castelnuovo’s handwritten notes for his series of lectures in the seminar for advanced students in 1927/28. (The other lecturers enumerated in the records of that seminar were Levi-Civita and Severi.) The word-by-words reproduction of the text is preceded by an introduction touching on both the historical and the mathematical aspects of these notes (a course on algebraic geometry given about 20 years after Castelnuovo had convinced himself that the Italian approach to algebraic geometry was insufficient to solve its main problems.) The notes give the theorems and facts presented, no proofs, which the speaker certainly could reproduce from memory. The emphasis is on construction of examples, both to show how past achievements were due to the study of examples and to impress future researchers that progress will be dependent on studies of examples. A central theme is Lüroth’s theorem, which shows that algebraic curves can be nicely treated based on rational curves and surfaces by rational and unirational ones, which fails for dimensions 3 and higher. The last section is a comment on later development after Castelnuovo, by Enriques (published by Conforto), Zariski, and their followers.

Reviewer: Heinrich Guggenheimer (Kfar Yona)