(Jurkiewicz--Danilov)
Let \(X_\Sigma\) be a smooth complete toric variety over \(\CC\) with rays \(\rho_1, \ldots, \rho_r\), primitive generators \(u_i\), and invariant divisors \(D_i\). Then \[ H^\bullet(X_\Sigma, \ZZ) \cong A^\bullet(X_\Sigma) \cong \ZZ[x_1, \ldots, x_r] / (I_\Sigma + J_\Sigma), \qquad x_i \mapsto [D_i] \in H^2, \] where \[ I_\Sigma = \big( x_{i_1} \cdots x_{i_k} \st \rho_{i_1}, \ldots, \rho_{i_k} \text{ do not span a cone of } \Sigma \big) \] is the Stanley–Reisner ideal of \(\Sigma\), and \(J_\Sigma = \big( \sum_i \inp{m}{u_i} x_i \st m \in M \big)\) is generated by the linear relations. In particular the odd cohomology vanishes and \(H^\bullet\) is generated in degree \(2\).
The Stanley–Reisner relation \(D_{i_1} \cdots D_{i_k} = 0\) says that invariant divisors whose rays span no cone have empty common intersection, since their orbit closures do not meet. The linear relations are \(\div(\chi^m) \sim 0\) from the class-group sequence.
For \(\PP^2\) with rays \(e_1, e_2, -e_1 - e_2\), the three rays together span no cone, so \(I_\Sigma = (x_1 x_2 x_3)\), and \(J_\Sigma = (x_1 - x_3, x_2 - x_3)\). So \(H^\bullet(\PP^2) = \ZZ[H]/(H^3)\). For the Hirzebruch surface \(\FF_a\) with rays \(e_1, e_2, -e_1 + a e_2, -e_2\), the opposite pairs span no cone, so \(x_1 x_3 = x_2 x_4 = 0\), with linear relations \(x_1 = x_3\) and \(x_2 = x_4 - a x_3\), and the ring is \(\ZZ[f, s]/(f^2, s^2 - a s f)\) for \(f = x_1\) and \(s = x_4\); so \(D_4^2 = a\) and \(D_2^2 = (s - af)^2 = -a\).
(Betti numbers)
Let \(X_\Sigma\) be a smooth complete toric variety of dimension \(n\), and let \(d_j\) be the number of \(j\)-dimensional cones of \(\Sigma\). Then the odd Betti numbers vanish and \[ \beta_{2k}(X_\Sigma) = \sum_{i=k}^{n} (-1)^{i-k} \binom{i}{k} d_{n-i} . \] In particular \(\chi(X_\Sigma) = d_n\), the number of maximal cones. [@Ful93]
For \(\PP^2\), \(d_0 = 1\) and \(d_1 = d_2 = 3\), so \(\beta_0 = d_2 - d_1 + d_0 = 1\), \(\beta_2 = d_1 - 2 d_0 = 1\) and \(\beta_4 = d_0 = 1\).
Dependencies
T-TORDIVDivisors, class group and Picard group from the raysuses