Lagmental Vicfred

Normalization Separates the Two Branches of a Node by Vicfred

tags: mathematics, smoothness, singularities, etale morphisms

The normalization of a nodal curve replaces the singular point by the distinct branches meeting there. The example is deliberately concrete; it is a test of the statement, not a substitute for it.

Statement

For \(X=V(f_1,\ldots,f_r)\subseteq\mathbf A^n\), the Jacobian matrix \(J_p=(\partial f_i/\partial x_j)(p)\) controls tangent dimensions. Smoothness asks for the expected rank after passing to the residue field.

$$ C=V(y^2-x^2(x+1)) $$

I read the first line as input and the second as output. The symbols \(\forall\) and \(\exists\) are not interchangeable, and neither may be upgraded silently to \(\Longleftrightarrow\).

$$ \nu:\widetilde C\to C\ \text{finite and birational} $$

Worked algebra

The middle display is intentionally dense: it is where signs, bounds, multiplicities, or normalising factors are most likely to be lost.

$$ x=t^2-1,\qquad y=t(t^2-1),\qquad\nu^{-1}(0,0)=\{t=1,\ t=-1\} $$

Conceptual compression

The aligned summary deliberately puts the datum and conclusion on different rows. Mathematically, this is the distinction between specifying an object and proving a property of it.

$$ \begin{aligned} \mathsf{D}\;&:\quad C=V(y^2-x^2(x+1)),\\[5pt] \mathsf{C}\;&:\quad \nu:\widetilde C\to C\ \text{finite and birational}. \end{aligned} $$

Caveat

A visually sharp point need not capture scheme-theoretic singularity, and characteristic can make every partial derivative vanish unexpectedly.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \nu:\widetilde C\to C\ \text{finite and birational} \end{gathered}} $$

With the dependency made explicit, the same pattern can be recognised safely in nearby problems. A changed hypothesis should now be easy to spot.

This article was posted on Tue 27 December 2011. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.