Write the invariant prime divisors in the order of the rays
$$
v_1=(1,0),\quad v_2=(0,1),\quad v_3=(-1,k),\quad v_4=(0,-1)
$$
as $D_1,\ldots,D_4$. The <toric divisor class sequence> is
$$
0\longrightarrow M\longrightarrow\mathbb Z^4
\longrightarrow\operatorname{Cl}(\mathbb F_k)\longrightarrow0,
$$
and the two characters in the standard basis of $M$ give
$$
D_1-D_3=0,
\qquad
D_2+kD_3-D_4=0.
$$
Thus
$$
\operatorname{Cl}(\mathbb F_k)
\cong\mathbb Z[S]\oplus\mathbb Z[F],
$$
where $S=D_2$, $F=D_1=D_3$, and $D_4=S+kF$. Since the fan is smooth, this is also the <Picard group>.

The <orbit-cone correspondence> decomposes $\mathbb F_k$ into one two-dimensional torus, four one-dimensional torus orbits, and four fixed points. Only the zero-dimensional orbits contribute to the compactly supported <Euler characteristic>, so
$$
\chi_{\mathrm{top}}(\mathbb F_k)=4.
$$

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 152 / 2 / a / Solution

For a toric variety whose rays span $N_{\mathbb R}$, the divisor class group is computed by
$$
0\longrightarrow M\longrightarrow\mathbb Z^{\Sigma(1)}
\longrightarrow\operatorname{Cl}(X_\Sigma)\longrightarrow0,
$$
where $m\mapsto(\langle m,v_\rho\rangle)_\rho$.


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 Incoming links: Toric divisor class sequence