Show that every normal affine curve over \(\CC\) is smooth.
Solution
Let \[ X=\mspec A \] be a normal affine curve over \(\CC\). Thus \(A\) is a finitely generated integral \(\CC\)-algebra of dimension \(1\) and is integrally closed in its fraction field.
1. For every point \(p\in X\), the local ring\[ \mco_{X,p}=A_{\mathfrak m_p} \]is a one-dimensional Noetherian normal local domain.
Because \(A\) is finitely generated over a field, it is Noetherian. Localization preserves the Noetherian property and the domain property.
Normality also localizes. Indeed, if \[ \frac ab\in\Frac(A) \] is integral over \(A_{\mathfrak m_p}\), then after clearing the finitely many denominators occurring in a monic integral equation, there is some \[ s\in A\setminus\mathfrak m_p \] such that \[ s\frac ab \] is integral over \(A\). Since \(A\) is integrally closed, \[ s\frac ab\in A, \] and hence \[ \frac ab\in A_{\mathfrak m_p}. \] Thus \(A_{\mathfrak m_p}\) is normal.
Finally, \(p\) is a closed point of the affine curve and \[ \dim X=1. \] The affine dimension theorem gives \[ \dim A_{\mathfrak m_p}=1. \]
2. Every local ring \(\mco_{X,p}\) is a discrete valuation ring and hence regular.
By step 1, \(\mco_{X,p}\) is a one-dimensional Noetherian normal local domain. By the one-dimensional normality criterion D-QJ5M9, such a ring is a discrete valuation ring. A DVR is a regular local ring of dimension \(1\).
3. Every point \(p\in X\) is smooth.
The field \(\CC\) is perfect. For a variety of finite type over a perfect field, a point is smooth exactly when its local ring is regular. Step 2 shows that \[ \mco_{X,p} \] is regular for every \(p\in X\). Hence every point is smooth.
4. Therefore\[ \boxed{X\text{ is smooth}.} \]
A variety is smooth when it is smooth at every point. This is step 3.
5. Q.E.D.
Steps <1>1–<1>4 prove that normality of an affine curve over \(\CC\) forces smoothness.