Bertini and his two fundamental theorems

TL;DR

This paper reviews Bertini's life, critically examines his two fundamental theorems, and provides complete, rigorous modern proofs, including a new extension of Bertini's first theorem to variable r-fold points.

Contribution

It offers a thorough historical and mathematical analysis of Bertini's theorems with new, rigorous proofs and an extension to r-fold points.

Findings

Provided complete, rigorous modern proofs of Bertini's theorems

Developed a new extension of Bertini's first theorem for variable r-fold points

Clarified the content of Bertini's theorems for non-specialists

Abstract

After reviewing Bertini's life story, a fascinating drama, we make a critical examination of the old statements and proofs of Bertini's two fundamental theorems, the theorem on variable singular points and the theorem on reducible linear systems. We explain the content of the statements in a way that is accessible to a nonspecialist, and we develop versions of the old proofs that are complete and rigorous by current standards. In particular, we prove a new extension of Bertini's first theorem, which treats variable $r$-fold points for any $r$.