Bertini and his two fundamental theorems

1997/04/17 by Steven L. Kleiman, Kleiman, Steven L. · 6 citations
Computer Science · Mathematics · #14C20 (Primary) 14-03 (Secondary) #Advanced Optimization Algorithms Research #Algebraic Geometry (math.AG) #FOS: Mathematics #Numerical Methods and Algorithms #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.alg-geom/9704018

openalex publication_date 1997/04/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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