Let X ⊆ ℙ^n be a smooth variety over an algebraically closed field. Bertini's theorem states that for a general hyperplane H ⊆ ℙ^n, the intersection X ∩ H is also smooth. In general, a "Bertini theorem" is a result like this, which states that some type of singularity is preserved under taking intersections with a general hyperplane, such as reduced, normal, Cohen-Macaulay, Gorenstein, and more.

We interpret these theorems in the language of commutative algebra (graded rings). Additionally, we discuss Bertini theorems for local rings and Bertini theorems for classes of singularities arising in the minimal model program.