Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 25F / Solution

For $X\subseteq {\mathbb{A}}^n$ and $P\in X$, the Zariski tangent space isThe dimension can be defined as ${\min }_{P\in X}\dim T_{P}X$. Equivalently, the krull dimension of an affine variety is the supremum of lengths of strict chains of irreducible closed subsets, or the Krull dimension of $k[X]$. A point is singular when $\dim T_{P}X>\dim X$.

Forthe differentials give the tangent spaces of a product of two nodal line pairsThe variety is a union of four two-dimensional linear spaces. If both pairs $(x_1,x_2)$ and $(x_3,x_4)$ are nonzero, the two displayed equations are independent and $\dim T_{P}X=2$. If exactly one pair is zero, the dimension is three; at the origin it is four. Thus the singular locus is the union of the loci where either coordinate pair vanishes.

For $Y$, rank at most one is equivalent to vanishing of all $2\times 2$ minors:These homogeneous equations prove that $Y$ is projectively Zariski closed.

The following affine charts show both its dimension and smoothness. On $y_0=1$,so $(y_1,y_2)$ are free. On $y_1=1$,so $(y_0,y_3)$ are free. On $y_3=1$,and on $y_4=1$,These four charts cover $Y$, since the point with only $y_2$ nonzero violates the second minor. Every chart is isomorphic to ${\mathbb{A}}^2$. Hence the smooth projective surface from overlapping rank-one coordinates satisfiesand is nonsingular everywhere.

Solved by gpt-5.6-sol high.