(The cone)
Take \(\sigma = \Cone(e_2,\ d e_1 - e_2) \subseteq N_\RR = \RR^2\) for \(d \geq 2\). Rotating as usual, \(\dualof{\sigma} = \Cone\big( (1,0),\ (1,d) \big)\), whose lattice points are generated by \[ (1,0), (1,1), \ldots, (1,d) \rightsquigarrow x, xy, \ldots, xy^d , \] so \(X_\sigma = \hat C_d\) is the affine cone over the rational normal curve of degree \(d\), cut out by the \(2 \times 2\) minors of \[ \begin{bmatrix} z_0 & z_1 & \cdots & z_{d-1} \\ z_1 & z_2 & \cdots & z_d \end{bmatrix} . \] For \(d = 2\) this is the quadric cone \(V(z_0 z_2 - z_1^2)\).
(The class group)
The two rays give \(D_1, D_2\), and the sequence \[ M \to \Div_T(U_\sigma) = \ZZ D_1 \oplus \ZZ D_2 \to \Cl(U_\sigma) \to 0 \] sends \(m \mapsto \sum_\rho \inp{m}{u_\rho} D_\rho\). On the basis \(e_1, e_2\) of \(M\) the matrix against the ray generators \(u_1 = e_2\), \(u_2 = d e_1 - e_2\) is \[ \begin{bmatrix} 0 & d \\ 1 & -1 \end{bmatrix} , \] that is \(\operatorname{div}(\chi^{e_1}) = d D_2\) and \(\operatorname{div}(\chi^{e_2}) = D_1 - D_2\). The second relation kills \(D_1\) in favour of \(D_2\), and the first says \(d D_2 = 0\): \[ \Cl(\hat C_d) = \ZZ/d\ZZ . \]
\(U_\sigma\) is affine, so \(\Pic(U_\sigma) = 0\) while \(\Cl(U_\sigma) = \ZZ/d\). Therefore \(D_2\) is a Weil divisor that is not Cartier — no case analysis needed, the two groups simply differ. This is the standard counterexample to “Weil implies Cartier” and, at \(d = 2\), the standard example of a normal surface singularity, \(V(xy - z^2)\) with its two rulings.
Dependencies
FE-TORDUALThe affine toric variety of the cone on (2,1) and (1,4)uses
Backlinks
FE-FULQUOTThe cone over the rational normal curve as a cyclic quotient singularityusesFE-TORMINRESThe minimal resolution of a toric surface singularityuses