Show that every normal projective curve over \(\CC\) is smooth.
Solution
Let \[ X\subseteq\PP^n_\CC \] be a normal projective curve, and let \[ U_i=\{x_i\ne0\}\subseteq\PP^n_\CC, \qquad X_i=X\intersect U_i. \] The nonempty \(X_i\) form an affine open cover of \(X\).
1. Every nonempty chart \(X_i\) is a normal affine curve over \(\CC\).
A projective variety is normal when each of its affine charts is normal, so each \(X_i\) is normal.
Since \(X\) is an irreducible projective curve, every nonempty open subset of \(X\) is irreducible and has the same dimension as \(X\). Hence every nonempty \(X_i\) has dimension \(1\). Thus each \(X_i\) is a normal affine curve over \(\CC\).
2. Every nonempty chart \(X_i\) is smooth.
Step 1 shows that \(X_i\) is a normal affine curve over \(\CC\). Every normal affine curve over \(\CC\) is smooth P-AGXVARNORMALCURVE, so \(X_i\) is smooth.
3. The projective curve \(X\) is smooth.
The charts \(X_i\) cover \(X\), and by step 2 every nonempty chart is smooth. Smoothness is local on the source, so every point of \(X\) is smooth. Hence \[ \boxed{X\text{ is smooth}.} \]
4. Q.E.D.
Steps <1>1–<1>3 reduce projective normality chartwise to the smoothness of normal affine curves.