Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 114 / 2 / c / Solution

With the indicated points as basepoints, $X$ is the smash product ${\mathbb{CP}}^2\wedge {\mathbb{CP}}^2$. The quotient map identifies its positive-degree cohomology with the relative cohomology of the product modulo the wedge, which under the Künneth isomorphism is the ideal generated by $xy$. Thusin degrees $4,6,6,8$. The only nonzero product of positive-degree basis elements isTogether with the unit in degree zero, this determines the cohomology ring.

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