Two letters by Guido Castelnuovo

TL;DR

This paper transcribes and explains two historical letters by Guido Castelnuovo discussing the pursuit of algebraic proofs for key properties of complex algebraic surfaces, highlighting developments leading to Deligne and Illusie's proof in the 1980s.

Contribution

It provides historical insights into Castelnuovo's correspondence and clarifies the mathematical ideas related to algebraic proofs of irregularity and regular 1-forms on algebraic surfaces.

Findings

Historical clarification of Castelnuovo's mathematical ideas

Explanation of the algebraic proof developments in the 1980s

Insight into the evolution of algebraic geometry concepts

Abstract

In this expository paper we transcribe two letters by Guido Castelnuovo, one to Francesco Severi, the other to Beniamino Segre, and explain the contents of both, which basically focus on the quest for an algebraic proof of the equality between the analytic and the arithmetic irregularity and of the closedness of regular 1-forms on a complex, projective, algebraic surface. Such an algebraic proof has been found only in the 1980's by Deligne and Illusie.