Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 2 / 24G (Algebraic Geometry)

In an affine chart, the Zariski tangent space at $P$ consists of vectors $v$ with $d{f}_P(v)=0$ for every polynomial vanishing on the variety. Intrinsically it is the dual of ${\mathfrak{m}}_P/{\mathfrak{m}}_P^2$. For a finite set of generators in an ambient $N$-space, its dimension is $N-\mathrm{rank}J(P)$. The condition that this be at least $r$ is the vanishing of all $(N-r+1)$-minors, hence is closed. The affine-chart descriptions agree intrinsically; being closed is local on the variety, so the same conclusion holds projectively.

For the two given quadrics the gradient rows areTheir intersection has dimension two: the quadrics have no common factor, so form a codimension-two complete intersection. Its singular points are where the two rows are dependent. If the first row vanishes, one obtains $[1:0:0:0:0]$. Otherwise write a dependence with nonzero coefficient of the second row. Its first coordinate forces $X_1=0$. The defining equations then imply $X_3{X}_4=0$ and $X_3^2+X_4^2=0$, hence $X_3=X_4=0$. Conversely every point of the resulting line lies on the variety and has dependent rows. ThereforeOn this line at least one of $X_0,X_2$ is nonzero, so the Jacobian rank is exactly one. The projective ambient tangent dimension is four, giving $\dim T_{V,P}=3$ at every singular point. Away from the line the rank is two and the tangent dimension is two.