Proof. Indeed, passing to the projectivization of these affine cones, $\mathbb {P}(\mathcal {S}q^{2}_f)$ is the second Veronese embedding of $\mathbb {P}(S^{1})$ in $\mathbb {P}(S^{2})$. Thus the positive generator $H$ of ${\rm Pic}(\mathbb {P}(\mathcal {S}q^{2}_f))$ satisfies the property that $H^{0}(\mathbb {P}(\mathcal {S}q^{2}_f), H)=:V$ has dimension $n+1$ and the restriction map $(S^{2})^{*}\rightarrow {\rm Sym}^{2}V$ is an isomorphism. The dual isomorphism gives the desired isomorphism ${\rm Sym}^{2}S^{1}\cong S^{2}$, with ${S^{1}:=V^{*}}$.

Proof. One has $f=\sum _j f_{j}$, where each $f_{j}$ involves variables $X_{j,1},\ldots, X^{j,i_j}$. It is immediate to check that the statement for each $f_{j}$ implies the statement for $f$. Turning to the $f_{j}$, they are either general linear sections of the quartic Pfaffian hypersurface in $\mathbb {P}^{27}$ by a $\mathbb {P}^{n}$, $n=23$ or $24$. Let us show that each of them has no infinitesimal automorphism. The automorphism group of the general Pfaffian hypersurface $\mathcal {P}f_k\subset \mathbb {P}\big (\bigwedge ^{2}V_{2k}\big )$ is the group ${\rm {P}GL}(2k)$. We claim that the automorphism group of a general linear section of dimension $>2(2k-2)={\rm dim}\,G(2,V_{2k})$ is also contained in ${\rm {P}GL}(2k)$. This follows from the fact that after blowing-up $\mathcal {P}f_{k}$ along its singular locus, which parameterizes forms of rank $<2k-2$, we get a dominant morphism $\widetilde {\mathcal {P}f_{k}}\rightarrow G(2,V_{2k})$, which to a degenerate form associates its kernel. If we consider a general linear section $X_l$ of $\mathcal {P}f_{2k}$ of dimension $>{\rm dim}\,G(2,V_{2k})$ defined by a $r$-dimensional vector subspace $W\subset \bigwedge ^{2}V_{2k}^{*}$, the same remains true and we get a morphism $\tilde {X}_l\rightarrow G(2,V_{2k})$ which is dominant with connected fiber of positive dimension. Thus the automorphism group of $X_l$ has to act on $G(2,V_{2k})$ and it has to identify with the group of automorphisms of $G(2,V_{2k})$, or automorphisms of $\mathbb {P}(V_{2k})$ preserving the space $W\subset \bigwedge ^{2}V_{2k}^{*}$. It is easy to check that this space is zero once $r\geq 3$. Coming back to our situation where $k=d=4$, our choices of $r$ are $r=3$ or $r=4$ for $k=4$. In all cases, the variety $X_l$ has dimension $>2(2k-2)$ so the analysis above applies.

Proof. Let as above $f=\sum _l f_{j}$. The singular locus $Z_f$ of $X_f=V(f)$ is the join of the singular loci $Z_{j}$ of $V(f_j)$ in $\mathbb {P}^{i_j}$. This means that, introducing the natural rational projection map $\pi :\mathbb {P}^{\sum _j i_j-1}\dashrightarrow \prod _l\mathbb {P}^{i_j-1}$, one has $Z_f=\pi ^{-1}\big (\prod _jZ_{j}\big )$.

Proof. Consider first the case of the Pfaffian linear sections $Z_j$. The claim follows in this case because they are general linear sections of the singular locus $Z$ of the quartic universal Pfaffian $\mathcal {P}f_4$ in $\mathbb {P}\big (\bigwedge ^{2} V_8\big )$, which is defined by the equations $\omega ^{3}=0$, that is, by cubics, and is not contained in any quadric. The last point can be seen by looking at the singular locus of $Z$, which consists of forms of rank $2$, that is the Grassmannian $G(2,V_8^{*})$. Along this locus, the Zariski tangent space of $Z$ is the full Zariski tangent space of $\mathbb {P}\big (\bigwedge ^{2}V_8\big )$. A quadric containing $Z$ should thus be singular along ${\rm Sing}\,Z$. But ${\rm Sing}\,Z=G(2,V_8^{*})$ is not contained in any proper linear subspace of $\mathbb {P}\big (\bigwedge ^{2} V_8\big )$. It follows that $Z$ is not contained in any quadric. It remains to conclude that the same statement is true for the general linear section $\mathbb {P}^{i_l-1}\cap \mathcal {P}f_4$, with $i_l=24,\ 25$. Its singular locus $Z_l$ is the general linear section $\mathbb {P}^{i_l-1}\cap Z$, and we show inductively that any quadric containing $Z_l$ is the restriction of a quadric containing $Z$. This statement only needs that $Z_l$ is nonempty (it has dimension $\geq 18$ in our case) and that all the successive linear sections $\mathbb {P}^{j}\cap Z$, with $j\geq i_l$, are linearly normal in $\mathbb {P}^{j}$, which is not hard to prove. Finally we have to show that the same is true for a general $f=\sum f_j$. As already mentioned, $Z_f$ is then a join $Z_1*\ldots *Z_l$ and a join of varieties not contained in any quadric is not contained in any quadric.

Proof. We know that $Z_f$ is the join of the $Z_j\subset \mathbb {P}^{i_j}$, where each $Z_j$ is a smooth linear section of the singular locus $Z$ of $\mathcal {P}f_4$ by either a $\mathbb {P}^{24}$ or a $\mathbb {P}^{23}$. Let us first conclude the proof when f is one of the $f_j$ (and thus Z is one of the $Z_j$). We observe that $Z\subset \mathbb {P}\big (\bigwedge ^{2}V_8\big )$ is the set of $2$-forms of rank $\leq 4$ (the generic element of $Z$ being of rank exactly $4$), and has a natural birational model $\tilde {Z}\rightarrow Z$, where

\[ \tilde{Z}\subset G(4,V_8^{*})\times \mathbb{P}\bigg(\bigwedge^{2}V_8\bigg),\quad \tilde{Z}=\{([W_4],\omega),\,W_4\subset {\rm Ker}\,\omega\}. \]

The statement (a) easily follows from the above description of $\tilde {Z}$ and the fact that the $Z_j$ are general linear sections of codimension $3$ or $4$ of $Z$. Let us prove (b). The variety $\tilde {Z}$ is smooth and, being a projective bundle fibration over $G(4,V_8^{*})$, has Picard rank $2$. Its effective cone is very easy to compute: indeed, the line bundle $l$ which is pulled-back from the Plücker line bundle on the Grassmannian via the first projection $pr_1$ is clearly one extremal ray of the effective cone since the corresponding morphism has positive dimensional fibers. There is a second extremal ray of the effective cone, which is the class of the divisor $D$ contracted by the birational map $\tilde {Z}\rightarrow Z$ (induced by the second projection $pr_2$). One easily computes that this class is $2h-l$, where $h$ is the pull-back of hyperplane class on $\mathbb {P}\big (\bigwedge ^{2}V_8\big )$ by $pr_2$. We observe that the fibers of $pr_1$ are of dimension $5$, so that, when we take a general linear section of $Z$ by a codimension $3$ or $4$ projective subspace, getting the singular locus $Z_j$ of $X_j$, all the properties above remain satisfied, and thus ${\rm Pic}\,\tilde {Z}_j=\mathbb {Z} h_j+\mathbb {Z}l_j$, with effective cone generated by $l_j$ and $2h_j-l_j$. We now finish the argument for $Z_j$: we lift our data $\{D_A\},\ D_0$ to $\tilde {Z}_j$. We then have

\[ 2h_j=\tilde{D}_0+2\tilde{D}_A \]

in ${\rm Pic}\,\tilde {Z}_j$, with $\tilde {D}_0$ effective and ${\rm dim}|\tilde {D}_A|\geq n$, where $n$ is ${\rm dim}|h_{\mid Z_j}|$ by (a). Let us write

\[ \tilde{D}_0=\alpha h_j+\beta l_j= \frac{\alpha}{2} (2h_j-l_j)+\bigg(\beta+ \frac{\alpha}{2}\bigg)l_j\quad {\rm in}\ {\rm Pic}\,\tilde{Z}_j, \]

with $\alpha,\beta \in 2\mathbb {Z}$. Then the analysis above shows that

\[ \alpha\geq 0,\quad \beta+ \frac{\alpha}{2}\geq 0. \]

As $2h_j-\tilde {D}_0$ is effective, we also have

\[ 2-\alpha\geq 0,\quad -\beta+1- \frac{\alpha}{2}\geq 0. \]

As $\alpha$ is even, we only get the possibilities $\alpha =0,\ 2$. If $\alpha =2$, we get $\beta =0$ and thus $2h_j-\tilde {D}_0=0$, contradicting the assumption ${\rm dim}|2h-\tilde {D}_0|\geq n$. If $\alpha =0$, we get from the inequalities above $\beta =0$ since $\beta$ has to be even, and $2h=2\tilde {D}_A$, which proves statement (b) (using the fact that ${\rm Pic}\,Z_j$ has no $2$-torsion and statement (a)).

We now have to prove the same result for the join $Z_f=Z_1*Z_2*\ldots * Z_l$, which is done inductively on the number $l$, assuming, as this is satisfied in our situation, that the $Z_i$ are simply connected. This way we are reduced to consider only the join $Z_1*Z_2\subset \mathbb {P}^{n}$, with $n=n_1+n_2+1$, of two linearly normal varieties $Z_1\subset \mathbb {P}^{n_1},\ Z_2\subset \mathbb {P}^{n_2}$, which satisfy the properties (a) and (b). We observe that $Z_1*Z_2$ is dominated by a $\mathbb {P}^{1}$-bundle over $Z_1\times Z_2$, namely

(32)\begin{equation} \widetilde{Z_1*Z_2}:=\mathbb{P}(\mathcal{O}_{Z_1}(1)\oplus \mathcal{O}_{Z_2}(1))\stackrel{\pi}{\rightarrow}Z_1\times Z_2, \end{equation}

the two sections being contracted to $Z_1$, respectively $Z_2$, by the natural morphism to $Z_1*Z_2\subset \mathbb {P}^{n}$. The description (32) of the join immediately proves (a) for $Z_1*Z_2$ once we have it for $Z_1$ and $Z_2$. We now turn to (b). Let $h=\mathcal {O}_{\mathbb {P}(\mathcal {O}_{Z_1}(1)\oplus \mathcal {O}_{Z_2}(1)})$ on $\widetilde {Z_1*Z_2}$ and let $D_0$ be a fixed effective divisor and $\{D_A\}$ be a mobile family of divisors on $\widetilde {Z_1*Z_2}$ such that

(33)$$\begin{gather} D_0+2D_A=2h, \end{gather}$$

(34)$$\begin{gather}{\rm dim}\{D_A\}\geq n. \end{gather}$$

Then either $D_0$ or $D_A$ is vertical for $\pi$. Indeed, they both restrict otherwise to a divisor of degree $\geq 1$ on the fibers of $\pi$, contradicting (33). Assume $D_0$ is vertical for $\pi$, that is, $D_0=\pi ^{-1}(D'_0)$. The equality $2h-D'_0=2D_A$ says that $D'_0=2D''_0$ as divisors on $Z_1\times Z_2$ and, as $Z_1$ and $Z_2$ are simply connected, $D''_0\in |pr_1^{*}D''_{0,1}+pr_2^{*}D''_{0,2}|$ and both $2D''_{0,1}, \ 2D''_{0,2}$ are effective. The divisors $D''_{0,i}$ on $Z_i$ have the property that the linear system $|h-pr_1^{*}D''_{0,1}-pr_2^{*}D''_{0,2}|$ on $\mathbb {P}(\mathcal {O}_{Z_1}(1)\oplus \mathcal {O}_{Z_2}(1))$ has dimension $\geq n$, which says that

\[ {\rm dim}|\mathcal{O}_{Z_1}(1)({-}D''_{0,1})|+{\rm dim}|\mathcal{O}_{Z_2}(1)({-}D''_{0,2})|\geq n_1+n_2. \]

As $2D''_{0,1}$ is effective on $Z_1$ and $2D''_{0,2}$ is effective on $Z_2$, we conclude that $D''_{0,i}=0$ and that the $D_A$ belong to $|\mathcal {O}_{Z_1*Z_2}(1)|$, so (b) is proved in this case.

In the case where $D_0$ is not vertical, then restricting again to the fibers of $\pi$, $D_0=2h-D'_0$ where $D'_0$ is effective and comes from $Z_1\times Z_2$, and $D_A$ is vertical, $D_A=\pi ^{-1}(D'_A)$. Hence we have again $D'_0=2D''_0$, and ${\rm dim}\,|h-D'_0-D'_A|\geq n$, so the proof finishes as before that $D'_0=0$ and $D_A=0$ which contradicts (34).

We now complete the proof of Lemma 2.4. We have $J_f\subset I_{Z_f}$, since $Z_f$ is contained in ${\rm Sing}\,X_f$. Let $T\subset S^{2}$ be a closed algebraic subset satisfying the assumptions of Lemma 2.4 for $f$. Then for any $A,B\in T$, $AB_{|Z_f}=M^{2}_{|Z_f}$, for some $M\in S^{2}$. This implies that the moving part of ${\rm div}\,A_{\mid Z_f}$ appears with multiplicity $2$. Hence ${\rm div}\,A_{\mid Z_f}=D_0+2D_A$. We now use Claim 2.5 which implies that the family of divisors ${\rm div}\,A_{\mid Z_f}$ is of dimension $\geq n$, hence also the family $\{D_A\}$ of divisors. We then conclude from Claim 2.6 that $T\subset \mathcal {S}q^{2}$.

Proof. This is a dimension count. The differential equations appearing in (39) are linear second-order equations determined by elements $U$ of ${\rm rank}\leq 4$ in ${\rm Sym}^{2}H^{0}(\mathbb {P}^{n},\mathcal {O}_{\mathbb {P}^{n}}(1))^{*}$. The nonzero elements $U$ of ${\rm rank}\leq 4$ are parameterized by a variety of dimension $4(n-3)+9$. Given a nonzero $U$, the differential equation $\partial ^{2}_U \phi =0$ determines a linear subspace $H_U$ of $H^{0}(\mathbb {P}^{n},\mathcal {O}_{\mathbb {P}^{n}}(d))$ of codimension $h^{0}(\mathbb {P}^{n},\mathcal {O}_{\mathbb {P}^{n}}(d-2))$ since

\[ \partial^{2}_U: H^{0}(\mathbb{P}^{n},\mathcal{O}_{\mathbb{P}^{n}}(d))\rightarrow H^{0}(\mathbb{P}^{n},\mathcal{O}_{\mathbb{P}^{n}}(d-2)) \]

is surjective. If (37) holds, the union of the spaces $H_U$ does not fill-in a Zariski open set of $H^{0}(\mathbb {P}^{n},\mathcal {O}_{\mathbb {P}^{n}}(d))$.

It follows that all the equations appearing in (39) are trivial, which says equivalently that for any $i,\ j$

\[ \frac{\partial}{\partial X_i}\frac{\partial}{\partial Y_j}- \frac{\partial}{\partial X_j}\frac{\partial}{\partial Y_i}=0\quad {\rm in}\ {\rm Sym}^{2}(H^{0}(\mathbb{P}^{n}, \mathcal{O}_{\mathbb{P}^{n}}(1))^{*}). \]

These equations exactly say that for some $\lambda \in \mathbb {C}$,

\[ \frac{\partial}{\partial Y_i}=\lambda \frac{\partial}{\partial X_i} \]

for $i=0,\ldots,n-1$. Thus $g-\lambda f$ satisfies $({\partial }/{\partial X_i})(g-\lambda f)=0$ for $i=0,\ldots,n-1$, and hence $g=\lambda f +\alpha X_n^{d}$ for some coefficient $\alpha$, which completes the proof.

Proof. The conditions on $d,\ n$ are used in the following.

Proof. The statement is equivalent to proving that, if $f$ is a generic homogeneous polynomial of degree $d$ in $n+1$ variables, $J_f^{d}\subset S^{d}$ does not contain any power $x^{d}$ of a linear form, or any sum $x^{d}-y^{d}$ of two such powers. In the first case, we get that $x\cdot x^{d-1}=0$ in $R^{d}_f$ and in the second case, we get that $(x-y)(x^{d-1}+x^{d-2}y+\cdots +y^{d-1})=0$ in $R^{d}_f$. By an easy dimension count, one sees that for generic $f$, the multiplication map $x:R_f^{d-1}\rightarrow R_f^{d}$ by any nonzero linear form $x\in S^{1}$ is injective, so in the first case we conclude that $x^{d-1}=0$ in $R_f^{d-1}$ and in the second case, $x^{d-1}+x^{d-2}y+\cdots +y^{d-1}=0$ in $R_f^{d-1}$, or equivalently

(40)\begin{equation} x^{d-1}\in J_f^{d-1}\quad {\rm or}\quad x^{d-1}+x^{d-2}y+\cdots+y^{d-1}\in J_f^{d-1}. \end{equation}

Using the fact that $n\geq 2$, and choosing a coordinate system such that $x=X_0,\,y=X_1$, we can write (40) as

(41)\begin{equation} X_0^{d-1}=\sum_i\alpha_i\frac{\partial f}{\partial X_i}\quad {\rm or}\quad X_0^{d-1}+X_0^{d-2}X_1+\cdots+X_1^{d-1}=\sum_i\alpha_i\frac{\partial f}{\partial X_i}\in J_f^{d-1} \end{equation}

for some nonzero coefficients $\alpha _i$. We thus get in both cases a nontrivial second-order equation

\[ \sum_i\alpha_i\frac{\partial^{2} f}{\partial X_2 \partial X_i}=0, \]

which is excluded by Lemma 3.5.

We now complete the proof. Using the lemma, the projectivized isomorphism $i$ induces an isomorphism $i_1:\mathbb {P}(S^{1})\cong \mathbb {P}(S^{1})$ between the two projected Veronese $v_f(\mathbb {P}(S^{1})) \subset \mathbb {P}(R^{d}_f)$ and $v_g(\mathbb {P}(S^{1})) \subset \mathbb {P}(R^{d}_g)$, that is, $i_1$ satisfies $i\circ v_f=v_g\circ i_1$. The projective isomorphism $i_1$ lifts to a linear isomorphism $\tilde {i}_1:S^{1}\cong S^{1}$. The incomplete Veronese embeddings $v_f$, respectively $v_g$ factor canonically through the complete Veronese embeddings

\[ \mathbb{P}^{n}(S^{1})\stackrel{V_d}{\rightarrow} \mathbb{P}(S^{d})\dashrightarrow \mathbb{P}(R^{d}_f), \]

(respectively $\mathbb {P}^{n}(S^{1})\stackrel {V_d}{\rightarrow } \mathbb {P}(S^{d})\dashrightarrow \mathbb {P}(R^{d}_f)$,) which implies that the following diagram,

(42)

where $\widetilde {i_d}:S^{d}\cong S^{d}$ is induced by $\widetilde {i_1}$, is commutative up to a scalar.

The vertical quotient maps have for respective kernels $J_f^{d}$, $J_g^{d}$. We thus conclude that ${\widetilde {i_d}(J_f^{d})=J_g^{d}}$, and thus, by Proposition 3.2, $X_f$ is isomorphic to $X_g$.