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  "body": "\n## Statement\n\nLet $k$ be an [[algebraically closed field|02SJG1HC32]] of characteristic $0$.\nLet $f \\colon X \\to Y$ be a morphism of [[varieties|HV6YBS0YDH]] over $k$, and assume that $X$ is nonsingular, that is, a [[regular scheme|ESPG0BJERZ]].\nThen there is a nonempty open subset $V \\subseteq Y$ such that the restriction $f^{-1}(V) \\to V$ is a [[smooth morphism|EXKGG9X6J0]].\n\n## Remarks\n\nThe proof reduces to the case where $f$ is dominant and then compares tangent spaces.\nIn characteristic $0$, the field extension $k(Y) \\subseteq k(X)$ is separable, so the map on differentials is generically surjective.\nThen $\\Omega_{X/Y}$ is locally free of the right rank over a dense open set, which gives smoothness there.\nFor example, the map $\\mathbb{A}^2_k \\to \\mathbb{A}^1_k$, $(x, y) \\mapsto x^2 + y^2$, is smooth over $V = \\mathbb{A}^1_k \\setminus \\set{0}$.\nIts fibre over $0$ is the pair of lines $x = \\pm i y$, which is singular at the origin.\nThe characteristic hypothesis is necessary.\nIn characteristic $p \u003e 0$, the Frobenius map $\\mathbb{A}^1_k \\to \\mathbb{A}^1_k$, $t \\mapsto t^p$, is a morphism of smooth curves.\nIts fibre over $a$ is $\\operatorname{Spec} k[t]/(t^p - a) = \\operatorname{Spec} k[t]/((t - b)^p)$ with $b^p = a$, a non-reduced point, so it is smooth over no nonempty open set.\n\n## Explanations\n\n### L1 human\n\nTake a smooth variety and a map from it to another variety.\nSome fibres of the map may be singular, but generic smoothness says that this only happens over a small closed set.\nOver almost every point of the target, the fibre is a smooth variety.\nIn positive characteristic this can fail, because a map like $t \\mapsto t^p$ has only thickened points as fibres.\n\n### L2 human\n\nGeneric smoothness is the algebraic counterpart of Sard's theorem on critical values of smooth maps.\nIt is the main tool behind Bertini's theorem: a general hyperplane section of a smooth projective variety in characteristic $0$ is smooth.\nIt also shows that, in characteristic $0$, in a family with smooth total space over a curve, only finitely many fibres are singular.\nIts proof uses that every finitely generated field extension in characteristic $0$ is separably generated.\n",
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