(The fan)
For weights \(q_0, \ldots, q_n\) with \(\gcd = 1\), choose \(v_0, \ldots, v_n \in N \cong \ZZ^n\) generating \(N\) and satisfying the single relation \[ q_0 v_0 + q_1 v_1 + \cdots + q_n v_n = 0 . \] The fan of \(\PP(q_0, \ldots, q_n)\) has these as rays, with maximal cones spanned by every \(n\) of them. Taking all \(q_i = 1\) and \(v_i = e_i\), \(v_0 = -\sum e_i\) recovers \(\PP^n\).
(P(1,1,2))
Solve \(1 \cdot v_0 + 1 \cdot v_1 + 2 \cdot v_2 = 0\) in \(\ZZ^2\) with \[ v_2 = (1,0), \quad v_1 = (0,1), \quad v_0 = (-2,-1) \] — check \((-2,-1) + (0,1) + 2(1,0) = (0,0)\), and \(v_1, v_2\) generate \(N\). The three maximal cones have determinants \[ \det(v_1, v_2) = -1, \qquad \det(v_2, v_0) = -1, \qquad \det(v_0, v_1) = -2 , \] so exactly one torus-fixed point is singular, an \(A_1\) point of index \(2\). Its cone is \(\Cone\big( (-2,-1), (0,1) \big)\), which is \(\Cone\big( (0,1), (2,-1) \big)\) after a change of basis: the quadric cone \(V(xy - z^2)\).
\(\Cl\big(\PP(1,1,2)\big) = \ZZ^3 / M \cong \ZZ\), and the classes of \(D_0, D_1, D_2\) are the weights \(1, 1, 2\). So \(\Cl\) is generated by \(\OO(1)\), which is a Weil divisor class that is not Cartier at the singular point; \(\Pic\) is the index-two subgroup generated by \(\OO(2)\). This is the smallest example where \(\Pic \subsetneq \Cl\) with both torsion free, and it is the cone over a conic in disguise.
(P(1,1,2) from a polygon)
Let \(P = \operatorname{Conv}(0, 2e_1, e_2) \subseteq M_\RR = \RR^2\), with lattice points \(0, e_1, 2e_1, e_2\). The characters of these points give \((\CC^*)^2 \to \PP^3\), \((s,t) \mapsto [1 : s : s^2 : t]\), and \(X_P\) is the closure of the image, \[ X_P = V(y_0 y_2 - y_1^2) \subseteq \PP^3 , \] the projective cone over a conic. The inward facet normals of \(P\) are \((0,1)\), \((1,0)\) and \((-1,-2)\), with the relation \(1 \cdot (1,0) + 2 \cdot (0,1) + 1 \cdot (-1,-2) = 0\), so the normal fan is the fan of \(\PP(1,1,2)\) above.
Dependencies
T-TORDIVDivisors, class group and Picard group from the raysuses