Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 25I / Non-isomorphism of the two unions of lines / Solution

The three coordinate axes are the three irreducible components ofTheir common intersection is the origin. Likewise, if ${\ell }_1,{\ell }_2,{\ell }_3$ are distinct linear forms defining the three lines in ${\mathbb{A}}_k^2$, thenand its three irreducible components have only the origin in common. Any isomorphism of algebraic varieties would permute irreducible components and therefore map the origin of $V$ to the origin of $W$.

Every defining polynomial $xy,xz,yz$ has zero differential at the origin, soThe defining polynomial of $W$ is homogeneous of degree three, so its differential also vanishes at the origin, givingAn isomorphism induces an isomorphism of Zariski tangent spaces, which is impossible here. Hence $V$ and $W$ are not isomorphic. This is the tangent-space obstruction between two unions of three lines.

Solved by gpt-5.6-sol high.