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A nonreduced equation can hide a smooth hypersurface

Example

In $C^{2}$ with coordinates $(x, y)$ , the equations $x = 0$ and $x^{2} = 0$ define the same complex-analytic hypersurface germ at the origin, namely the smooth germ of the line ${x = 0}$ (Complex-analytic hypersurface germ and its reduced equation). The equation $x^{2}$ is not reduced, and its raw differential $d (x^{2}) = 2 x d x$ vanishes at every point of the hypersurface, whereas the differential $d x$ of the reduced equation $x$ never vanishes. This is why the regularity criterion is stated for a reduced local equation (Regular and singular points of an analytic hypersurface).

Facts & Assumptions

Given: The two equation germs $x$ and $x^{2}$ at $0 \in C^{2}$ and their common zero germ $X = Z (x) = Z (x^{2})$ .

[F1]

The coordinate germ $x$ is irreducible in $O_{C^{2}, 0}$ : it lies outside $m_{0}^{2}$ because its linear part is nonzero, while a product of two nonunits lies in $m_{0}^{2}$ ; hence $x$ is not a product of two nonunits (Irreducible and prime elements of an integral domain).

[F2]

An irreducible germ is reduced, because a germ divisible by the square of an irreducible germ is a product of two nonunits; the square-free reduction $g_{r e d}$ of a nonzero nonunit $g$ is reduced, depends on $g$ only up to associates, and satisfies $Z (g_{r e d}) = Z (g)$ on a common neighbourhood (Reduced holomorphic germ for a hypersurface, Square-free reduction of a holomorphic equation).

[F3]

A hypersurface germ is determined by its reduced defining germ, which is unique up to a unit; two defining equations give the same hypersurface germ exactly when their square-free reductions are associates (Complex-analytic hypersurface germ and its reduced equation).

[F4]

A point $q \in X$ is regular exactly when the differential of a local reduced equation of $X$ at $q$ is nonzero; equivalently, exactly when $X$ is a holomorphic hypersurface graph near $q$ (Regular and singular points of an analytic hypersurface).

Proof technique: direct — compute the square-free reductions and compare the two differentials on the common zero set.

Verification

1.1givenF1F2F3

The germ $x$ is irreducible by [F1] and hence reduced by [F2]; its factorisation has the single irreducible factor $x$ , so the square-free reduction of $x^{2}$ is $x$ and the square-free reduction of $x$ is itself. By [F2] we have $Z (x) = Z (x^{2})$ near $0$ , so the two equations define the same hypersurface germ $X$ , with reduced defining germ $x$ by [F3]; the equation $x^{2}$ is not reduced, because the irreducible germ $x$ divides it twice.

2.1step 1.1F4construct

The germ $X = {x = 0}$ is the graph $X = {(x, y) : x = 0}$ of the zero function over the $y$ -coordinate, hence is a holomorphic hypersurface graph near each of its points; by [F4] every point of $X$ is regular, and $X$ is smooth.

3.1step 1.1step 2.1F3F4algebra∎

On the one hand $d x$ is the constant nonzero covector $d x$ , so the differential of the reduced equation $x$ never vanishes and the criterion [F4] is satisfied at every point of $X$ . On the other hand $d (x^{2}) = 2 x d x$ vanishes at every point of $X$ , because $x = 0$ there. Thus the raw differential of the nonreduced equation $x^{2}$ vanishes on the very hypersurface on which the reduced equation $x$ has nonzero differential, and the regularity criterion must specify a reduced equation.

Depends on

Used by

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Sources

Jiří Lebl, Tasty Bits of Several Complex Variables, Chapter 6 §§6.1–6.7 (standard reference, not scraped)
Jean-Pierre Demailly, Complex Analytic and Differential Geometry, Chapter II §§2, 4 and 6 (standard reference, not scraped)