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We have an expression m g /1 = in OP,Ank , where ai,j ∈ A(Ank ), and bi,j ∈ then yield (ii). ai,j /bi,j fi i=1 A(Ank ) nP . Setting f to be the product of the fi,j will We rephrase the second part of the lemma more geometrically as follows. 2. If P ∈ X ⊆ Ank is a nonsingular point of an algebraic set, with dimP X = d, and if we choose f1 , . . , fn−d ∈ I(X) so that J(f1 , . . , fn−d )(P ) has rank n − d, then there exists f ∈ A(Ank ) such that f (P ) = 0, and I(X Z(f )) = (f1 , . . , fn−d ).

This fits with intuition – a point where two components intersect should not be nonsingular – but it’s not so obvious from either definition. 5. 11. If X is an affine algebraic set, then the set of singular points of X is a nowhere dense closed subset of X. Proof. Let Z ⊆ X be the points which are contained in more than one irreducible component of X; then Z is closed. If U = X Z, then U is open and dense in X, and is a disjoint union of components U1 , . . , Un whose closures Z1 , . . , Zn are the irreducible components of X.

One can vary the definition a bit by defining a notion of equivalence of atlases and speaking of a prevariety as a set with an equivalence class of atlases, or alternatively, by requiring an atlas to be maximal. Either of these options removes the “dependence on choice” of the atlas, but at this 39 point it is not clear whether it would be any less technical to simply do what modern algebraic geometers do, which is to work with sheaves. 3. Any affine variety “is” a prevariety, with an atlas consisting of a single chart.

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Algebraic Varieties by Brian Osserman

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