Affine Perverse Vanishing And Universal Hyperplanes Examples

The source ${\mathbb{P}}^n$ is smooth and irreducible of dimension $n$, the only fibre of $f$ is all of ${\mathbb{P}}^n$ of dimension $n$, and the image is a point of dimension zero, so the defect formula $r(f)={\max }_{Y_i\mathrm{\ne }\mathrm{\varnothing }}(2i+\dim Y_i-n)$ inline in General projective hyperplanes decrease the defect of semismallness gives $r(f)=2n-n=n$. The universal incidence $\mathcal{X}=\{(x,H):x\in H\}\subseteq {\mathbb{P}}^n\times ({\mathbb{P}}^n{)}^{\vee }$ is the projective-space bundle of relative dimension $n-1$ over the dual space $({\mathbb{P}}^n{)}^{\vee }$, since its fibre over a hyperplane $H$ consists of the points of $H\cong {\mathbb{P}}^{n-1}$, and on each standard dual chart a nonzero coefficient of the hyperplane equation eliminates one coordinate, giving a local trivialization; hence its source dimension is $n+(n-1)=2n-1$.

The universal family $g:\mathcal{X}\to ({\mathbb{P}}^n{)}^{\vee }$ has the single fibre dimension $n-1$ over the whole dual space, so its defect is $2(n-1)+n-(2n-1)=n-1$, strictly below $n$. This directly checks the positive-defect clause of General projective hyperplanes decrease the defect of semismallness, including $n=1$, where the incidence is the graph of an isomorphism onto the dual projective line and the defect is zero, as the formula gives.