Bertini's theorem
= Bertini's theorem
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= Bertini theorem
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For a smooth integral projective variety over $\mathbb C$, a general hyperplane section in a projective embedding is smooth; if its dimension is at least two, the section is also irreducible. More general statements concern linear systems and their base loci. The smoothness statement is developed in https://stacks.math.columbia.edu/tag/0FD4[the Stacks Project's Bertini section].