Bertini and his two fundamental theorems

Details Bertini and his two fundamental theorems Bertini and his two fundamental theorems

1997-04-17

arxiv.org/abs/alg-geom/9704018v1

Mathematics

Algebraic Geometry

PLAIN TeX, 29 pages with 3 figures, automatically produced by dvips

Scientific paper

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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