Given a compact toric manifold $X$ with Picard number $r$ it is well known that the cohomology ring $H^*(X;\mathbb{C})$ is a quotient of the polynomial ring $\mathbb{C}[p_1,\dots,p_r]$ by the ideal $I$ of the so called Kirwan's relations. If $r\leq 2$, then it is easy to see that the minimal number of generators of $I$ is $r$. Are there examples in which the minimal number of generators is not $r$?
1 Answer
There are counterexamples when $r$ equals $3$. Let $X$ be the blowing up of $\mathbb{P}^2$ in two distinct, torus invariant points. Let $p_1$ denote the pullback to $X$ of $c_1(\mathcal{O}_{\mathbb{P}^2}(1))$. Let $p_2$ denote the first Chern class of the invertible sheaf associated to the first exceptional divisor. Let $p_3$ denote the first Chern class of the invertible sheaf associated to the second exceptional divisor. Then $I$ is generated by the five relations $$g_1 = p_1\cdot p_2, \ g_2 = p_1\cdot p_3, \ g_3 = p_2\cdot p_3,\ g_4 = p_2^2 + p_1^2,\ g_5 = p_3^2+p_1^2.$$