Universal hyperplanes give relative weak Lefschetz and maximal pullback subobjects

By [F2], $p^{\mathrm{\prime }}{}^{\ast }K[d]$ is perverse and $i^{\ast }[-1]$ is perverse-exact on these product strata, so $M=i^{\ast }{p}^{\mathrm{\prime }}{}^{\ast }K[d-1]$ is perverse. Let $j:X^{\mathrm{\prime }}\setminus \mathcal{X}\hookrightarrow X^{\mathrm{\prime }}$ and $u=(f\times 1)j$. Over an affine chart of the hyperplane parameter space the complement of the incidence in ${\mathbb{P}}^d$ is an affine $d$-space bundle, and $X^{\mathrm{\prime }}\setminus \mathcal{X}$ is closed in the corresponding bundle over $Y^{\mathrm{\prime }}$; thus $u$ is affine. Apply $R(f\times 1{)}_{!}$ to the localization triangle $j_{!}{j}^{\ast }{p}^{\mathrm{\prime }\ast }K[d]\to p^{\mathrm{\prime }\ast }K[d]\to i_{\ast }{i}^{\ast }{p}^{\mathrm{\prime }\ast }K[d]\to$. By [F1] the first term lies in the perverse lower aisle, so it has no negative perverse cohomology; by proper base change and [F2] the second is $p^{\ast }R{f}_{\ast }K[d]$ and the third is $R{g}_{\ast }M[1]$. The long exact perverse sequence gives the restriction isomorphisms and the monomorphism. Dualizing the whole construction and using $D{i}_{\ast }=i_{!}D$, $i^{!}{K}^{\mathrm{\prime }}=i^{\ast }{K}^{\mathrm{\prime }}[-2]$ from [F2] gives the Gysin isomorphisms and the epimorphism for $l=1$.

The projective-space cohomology formula gives $R{p}_{\ast }{p}^{\ast }T={⨁}_{a=0}^{d}T[-2a]$ for every constructible complex $T$, and the identical formula with $d-1$ holds for $p^{\mathrm{\prime }}i:\mathcal{X}\to X$. To justify the formula as a morphism and not only stalk ranks, cup powers of the tautological Euler class pulled back from $({\mathbb{P}}^d{)}^{\vee }$ define the map from the direct sum, and on every fibre its map is the usual projective-space cohomology basis, hence an isomorphism by proper base change; the integral power basis is The complex tautological Euler class restricts to the projective-fiber generator, and tensoring the finite cellular cochain complex with $k$ preserves the basis by Cellular cochains compute cohomology with local coefficients. For heart objects $A,B$, adjunction therefore identifies $\mathrm{Hom}(p^{\ast }A[d],p^{\ast }B[d])$ with $\mathrm{Hom}(A,B)$: all terms $\mathrm{Hom}(A,B[-2a])$ for $a>0$ vanish by [F3].

Write $F=p^{\ast }[d]$. For the product projection $p$, the normal slices to $S\times {\mathbb{P}}^d$ are precisely the normal slices to $S$. Thus the stalk and costalk boundary conditions characterizing intermediate extension in [F3] are preserved: on a boundary stratum of dimension $s$ their strict bounds $q<-s$ and $q>-s$ become $q<-(s+d)$ and $q>-(s+d)$ after the shift $[d]$. On the dense open, pullback is the shifted pulled-back local system. The no-boundary characterization in [F3] therefore shows that $F$ takes ${\mathrm{IC}}_S(L)$ to ${\mathrm{IC}}_{S\times {\mathbb{P}}^d}(p^{\ast }L)$, with both intersection complexes normalized by their respective support dimensions. The local system $p^{\ast }L$ remains irreducible: a sub-local-system restricts along the section $y\mapsto (y,H_0)$ to a sub-local-system of $L$, hence to zero or all of $L$. Its rank is constant on each connected projective-space fibre (any two points lie on a projective line), so those two alternatives give zero or all of $p^{\ast }L$ on the whole product. Hence $F$ is exact, fully faithful by step 1.2, and preserves simples. Every composition factor of $F(A)$ is therefore a pulled-back simple; for a subobject $B\subseteq F(A)$ choose a simple $F(S)\subseteq B$, lift its inclusion to $S\subseteq A$ by full faithfulness, and induct on $B\mathrm{/}F(S)\subseteq F(A\mathrm{/}S)$. The lifted inclusion of the resulting base object into $A\mathrm{/}S$ has a preimage in $A$, whose exact image under $F$ is $B$. This proves the image of $F$ is closed under subobjects and quotients. The derived right adjoint $R{p}_{\ast }[-d]$ preserves the lower aisle: for $E\in {}^p{D}^{\ge 0}$ and $B\in {}^p{D}^{\le -1}$, adjunction gives $\mathrm{Hom}(B,R{p}_{\ast }E[-d])=\mathrm{Hom}(F(B),E)=0$. Thus $R{p}_{\ast }$ sends ${}^p{D}^{\ge q}$ into ${}^p{D}^{\ge q-d}$. Its heart right adjoint is $r={}^p{H}^{-d}R{p}_{\ast }$, since the positive-degree truncation of $R{p}_{\ast }C[-d]$ has no Hom from a heart object. The analogous left adjoint is $\mathrm{\ell }={}^p{H}^{d}R{p}_{!}$, using $p^{!}=p^{\ast }[2d]$. Full faithfulness gives $rF=1$. The kernel of the counit $Fr(C)\to C$ lies in the image of $F$ by subobject closure, and applying the left-exact $r$ shows that kernel is zero, by the adjunction identity $r({\epsilon }_C)=1$. Every subobject $F(B)\hookrightarrow C$ factors through this counit by adjunction. Consequently $Fr(C)$ is exactly the largest subobject coming from $Y$; duality gives the largest quotient $F\mathrm{\ell }(C)$.

Put $T=R{f}_{\ast }K$, $V=R(f\times 1{)}_{\ast }{p}^{\mathrm{\prime }\ast }K$ and $U=R{g}_{\ast }{i}^{\ast }{p}^{\mathrm{\prime }\ast }K$. The graph of $(f,X\hookrightarrow {\mathbb{P}}^d)$ is a closed embedding $a:X\hookrightarrow Y\times {\mathbb{P}}^d$ and $f=qa$, where $q$ is the projection. Closed pushforward is perverse-exact directly from the two stratum inequalities in [F3]: on strata inside the image its stalks and costalks are those of $K$, and on the complement they vanish. Applying the lower-aisle calculation of step 2.1 to $q$ therefore gives $T=R{q}_{\ast }{a}_{\ast }K\in {}^p{D}^{\ge -d}$. Under the cup-power bases of step 1.2, the map $R{p}_{\ast }V\to R{p}_{\ast }U$ is a matrix $(1,\beta ):{⨁}_{a=0}^{d-1}T[-2a]\oplus T[-2d]\to {⨁}_{a=0}^{d-1}T[-2a]$: restriction carries each of the first $d$ parameter-class powers to the identical power on the incidence bundle, while the last power may be a linear combination involving base classes. Changing coordinates by $(x,z)\mapsto (x+\beta z,z)$ identifies this map with projection. Its discarded summand $T[-2d]$ lies in ${}^p{D}^{\ge d}$, so the map induces isomorphisms in every perverse degree below $d$, including degrees $-1$ and $0$ for all $d\ge 1$. This uses no decomposition of $T$ into its perverse cohomology.

Set $n=d-1$. Step 1.1 gives an isomorphism $L={}^p{\tau }_{<n}V\simeq {}^p{\tau }_{<n}U$ and a monomorphism $\iota :{}^p{H}^{n}V\hookrightarrow {}^p{H}^{n}U$. Apply $R{p}_{\ast }$ to the two triangles $L\to V\to {}^p{\tau }_{\ge n}V\to$ and $L\to U\to {}^p{\tau }_{\ge n}U\to$. Their five-term exact sequences are ${}^p{H}^{-1}R{p}_{\ast }L\to {}^p{H}^{-1}R{p}_{\ast }V\to {}^p{H}^{-1}R{p}_{\ast }{}^p{\tau }_{\ge n}V\to {}^p{H}^{0}R{p}_{\ast }L\to {}^p{H}^{0}R{p}_{\ast }V$ and the same sequence for $U$. The first and fourth vertical maps are identities, and the second and fifth are isomorphisms by step 3.1; the five lemma proves that the middle map is an isomorphism. For either $W=V$ or $W=U$, the triangle $({}^p{H}^{n}W)[-n]\to {}^p{\tau }_{\ge n}W\to {}^p{\tau }_{\ge n+1}W\to$ has last term whose $R{p}_{\ast }$ lies in ${}^p{D}^{\ge n+1-d}={}^p{D}^{\ge 0}$, by step 2.1. Its perverse cohomology in degrees $-2$ and $-1$ vanishes. The resulting exact sequence gives, naturally in $W$, ${}^p{H}^{-1}(R{p}_{\ast }{}^p{\tau }_{\ge n}W)\simeq {}^p{H}^{-1-n}(R{p}_{\ast }{}^p{H}^{n}W)=r({}^p{H}^{n}W)$, since $-1-n=-d$. Thus $r(\iota )$ is an isomorphism. Proper base change and exact $F$ give ${}^p{H}^{n}V=F({}^p{H}^{-1}T)$, so its right adjoint is ${}^p{H}^{-1}T$. Adjunction factors $\iota$ as the counit $Fr({}^p{H}^{n}U)\to {}^p{H}^{n}U$ preceded by the isomorphism $F(r(\iota ))$. By step 2.1 its image is exactly the largest pullback subobject of ${}^p{H}^{n}U={}^p{H}^0(R{g}_{\ast }M)$. Apply this result to $DK$ and dualize using [F2] and [F3]; the middle object is dualized, perverse degrees change sign, and largest subobject becomes largest quotient $F({}^p{H}^{1}T)$. This proves every assertion.