$\nabla{F}$, its image, and its ramification. For diagonal $F$, a more
explicit analysis is possible (see §2.2).

###  A.1. The dual hypersurface and the discriminant form

Since $V$ is smooth, the _polar map_
$[\nabla{F}]\colon\mathbb{P}^{m-1}\to\mathbb{P}^{m-1}$ is _regular_ , i.e.
defined everywhere. (It would be better to write
$\mathbb{P}^{m-1}\to(\mathbb{P}^{m-1})^{\vee}$, but no confusion should
arise.) The map $[\nabla{F}]$ is finite of degree $2^{m-1}$
[dolgachev2012classical]*p. 29 between irreducible equidimensional projective
varieties, hence surjective. Since $\mathbb{P}^{m-1}$ is smooth, $[\nabla{F}]$
must then be flat (by “miracle flatness”).

Upon restricting $[\nabla{F}]$ to $V$, we get the finite surjective _Gauss
map_ $\gamma\colon V\to V^{\vee}$, where
$V^{\vee}\subseteq(\mathbb{P}^{m-1}_{\mathbb{Q}})^{\vee}$ denotes the _dual
variety_ of $V$, i.e. the closure of the union of hyperplanes tangent to the
_smooth locus_ $V_{\operatorname{sm}}$ of $V$. (For us, $V$ is pure of
codimension $1$, and $V_{\operatorname{sm}}=V$. So
$V^{\vee}=\overline{[\nabla{F}]V_{\operatorname{sm}}}=\overline{[\nabla{F}]V}=[\nabla{F}]V$.
Hence $\gamma$ is indeed well-defined and surjective; and $\gamma$ is finite
because it is a quasi-finite map between projective varieties.)

Here $V/\mathbb{Q}$ is irreducible over $\mathbb{C}$, so $V^{\vee}$ must be
too, by [gelfand2008discriminants]*p. 15, Proposition 1.3. Thus
$V^{\vee}/\mathbb{Q}$ is a projective _hypersurface_ , cut out by an
_absolutely irreducible form_ $F^{\vee}\in\mathbb{Q}[\bm{c}]$. (At least for
diagonal $F$, one can easily explicitly compute $F^{\vee}$; see §2.2.)

######  Remark A.1.

The notation $F^{\vee}$ is convenient for us, but likely _not_ standard.

######  Proposition A.2 (Classical).

Here $\deg{F^{\vee}}=3\cdot 2^{m-2}$. Given
$\bm{c}\in\mathbb{C}^{m}\setminus\\{\bm{0}\\}$, we have $F^{\vee}(\bm{c})=0$
if and only if $(V_{\mathbb{C}})_{\bm{c}}$ is singular. Furthermore, we may
choose $F^{\vee}\in\mathbb{Z}[\bm{c}]$ so that for all
$\bm{c}\in\mathbb{Z}^{m}$ and primes $p\nmid F^{\vee}(\bm{c})$, the special
fiber $(\mathcal{V}_{\bm{c}})_{\mathbb{F}_{p}}$ is smooth of dimension $m-3$.

######  Remark A.3.

Proposition A.2, together with the absolute irreducibility of $F^{\vee}$,
implies that [wang2021_large_sieve_diagonal_cubic_forms]*§1.2, Proposition-
Definition 1.8 is valid for all $F$ smooth over $\mathbb{Q}$—even though
[wang2021_large_sieve_diagonal_cubic_forms] only provides a proof when $F$ is
diagonal.

###### Proof sketch for Proposition A.2.

For the degree computation, see e.g. [dolgachev2012classical]*p. 33, (1.47).

For the “singular hyperplane section” interpretation over $K=\mathbb{C}$ or
over $K=\mathbb{F}_{p}$ with $p>p_{0}$, see e.g. [hooley2014octonary]*pp.
244–245. (The key is that _if_ $\mathcal{V}_{K}$ is smooth and $\bm{c}\in
K^{m}\setminus\\{\bm{0}\\}$, then $(\mathcal{V}_{K})_{\bm{c}}$ is singular if
and only if there exists $[\bm{x}]\in(\mathcal{V}_{K})_{\bm{c}}(\overline{K})$
such that $[\bm{c}]=[\nabla{F}]([\bm{x}])$.) We can then “absorb” the primes
$p\leq p_{0}$ (once and for all) into the polynomial $F^{\vee}$.

Alternatively, for a convenient reference touching briefly on both matters
above (at least in characteristic $0$), see [browning2009quantitative]*pp.
115–116, §7.1. ∎

It is known that $(V^{\vee})^{\vee}=V$ [dolgachev2012classical]*p. 30,
Reflexivity Theorem. Furthermore, the definition of $V^{\vee}$ (together with
the fact that $V,V^{\vee}$ are hypersurfaces with $V$ irreducible) implies the
divisibility $F(\bm{x})\mid F^{\vee}(\nabla{F}(\bm{x}))$, and reflexivity
(together with the fact that $V^{\vee},V$ are hypersurfaces with $V^{\vee}$
irreducible) similarly implies the divisibility $F^{\vee}(\bm{c})\mid
F(\nabla{F^{\vee}}(\bm{c}))$. These (symmetric!) divisibilities capture much
of the basic duality theory for $V$.

The apparent symmetry between $V,V^{\vee}$ thus far is deceptive, however.
What complicates matters is that $V^{\vee}$, or equivalently $F^{\vee}$, must
be _singular_ if $\deg{F}\geq 3$. (Otherwise, $(V^{\vee})^{\vee}$ would be a
hypersurface of degree larger than $\deg{F}$, contradicting reflexivity.)
Hence the polar map
$[\nabla{F^{\vee}}]\colon\mathbb{P}^{m-1}\dashrightarrow\mathbb{P}^{m-1}$ is
only a _rational_ map, defined away from $\operatorname{Sing}(V^{\vee})$ (a
proper closed subset of $V^{\vee}$).

Nonetheless, given _smooth_ points $[\bm{x}]\in V$ and $[\bm{c}]\in V^{\vee}$,
the biduality theorem says that
$[\nabla{F}(\bm{x})]=[\bm{c}]\iff[\nabla{F^{\vee}}(\bm{c})]=[\bm{x}]$. (See
e.g. [dolgachev2012classical]*p. 30, sentence after Reflexivity Theorem, which
however does not explicitly refer to the biduality theorem by name.) For us,
$V_{\operatorname{sm}}=V$, so biduality implies that
$[\nabla{F}],[\nabla{F^{\vee}}]$ restrict to _inverse morphisms_ between
$V\setminus[\nabla{F}]^{-1}(\operatorname{Sing}(V^{\vee}))$ and
$V^{\vee}\setminus\operatorname{Sing}(V^{\vee})$.

######  Remark A.4.

The map $\gamma\colon V\to V^{\vee}$ is finite surjective (and birational,
i.e. of degree $1$), but not necessarily _flat_ (or equivalently, _locally
free_). In fact, the finite $\mathcal{O}_{V^{\vee}}$-algebra
$\gamma_{\ast}\mathcal{O}_{V}$ is isomorphic to $\mathcal{O}_{V^{\vee}}$
_generically_ over $V^{\vee}$, but _not necessarily everywhere_ —for instance,
the geometric fiber
$V\times_{\gamma}\overline{k(p)}=\operatorname{Spec}((\gamma_{\ast}\mathcal{O}_{V})_{p}\otimes_{\mathcal{O}_{V^{\vee},p}}\overline{k(p)})$,
and in fact the analogous set-theoretic geometric fiber, may have size $\geq
2$ at some point $p\in\operatorname{Sing}(V^{\vee})$.

######  Question A.5.

What is known about the fiber of $\gamma$ over a _singular_ point $[\bm{c}]\in
V^{\vee}$?

Presumably the singular structure of $V_{\bm{c}}$ should play a role. Perhaps
works of Aluffi and Cukierman (such as [aluffi1993multiplicities,
aluffi1995singular]) can help to give a precise statement.

###  A.2. Ramification and the Hessian

$[\nabla{F}]$ is a finite surjective morphism of smooth varieties, so its
ramification theory is well-behaved. By [dolgachev2012classical]*p. 29,
Proposition 1.2.1,

  1. (1)

the _ramification divisor_ of
$[\nabla{F}]\colon\mathbb{P}^{m-1}\to\mathbb{P}^{m-1}$ is the Hessian
hypersurface $\operatorname{hess}(V)\subseteq\mathbb{P}^{m-1}$; and

  2. (2)

its image, the _branch divisor_ , is $\operatorname{St}(V)^{\vee}$, the dual
of the Steinerian hypersurface $\operatorname{St}(V)$ (with
$\operatorname{St}(V)$ defined scheme-theoretically as in
[dolgachev2012classical]*p. 19, §1.1.6).

In particular, both $\operatorname{hess}(V),\operatorname{St}(V)^{\vee}$ are
(possibly reducible or non-reduced) projective hypersurfaces in
$\mathbb{P}^{m-1}$.

When studying the Hasse principle (or even weak approximation), one can
certainly localize before counting points. Thus a useful fact worth mentioning
now is the following:

######  Proposition A.6 ([hooley1988nonary]*p. 37, Lemma 1).

If $V$ is a smooth cubic, then $V\not\subseteq\operatorname{hess}(V)$.

######  Remark A.7.

Thus on an _arbitrary_ smooth cubic $V$, as in [hooley1988nonary] or
[hooley2014octonary] for instance, it is always _non-vacuous_ to count points
on $V$ with nonvanishing Hessian determinant.

######  Remark A.8 (Cf. [hooley1988nonary]*p. 37, remarks in paragraph before
Lemma 1).

The intersection $V\cap\operatorname{hess}(V)$ consists of _inflection points_
if $m=3$, and of _parabolic points_ if $m\geq 4$ [dolgachev2012classical]*p.
17, Theorem 1.1.20. But it does not seem easy to find a general reference
proving the existence of non-inflection or non-parabolic points on $V$
(according as $m=3$ or $m\geq 4$).

The following stronger technical question comes up in Proposition 2.2,
although we happen to be able to sidestep it there (as remarked after the
proof of Proposition 2.2).

######  Question A.9.

Is it always true (for smooth cubic $V$) that
$V^{\vee}\not\subseteq\operatorname{St}(V)^{\vee}$?

######  Remark A.10.

If $V\subseteq\operatorname{hess}(V)$, then applying $[\nabla{F}]$ would imply
$V^{\vee}\subseteq\operatorname{St}(V)^{\vee}$. Thus an affirmative answer to
the question would give another proof of Proposition A.6.

##  Appendix B Isolating the singular series

######  Lemma B.1.

If $m\geq 4$, then $\sum_{n\asymp N}n^{1-m/2}\max_{n_{\star}\mid
n}n_{\star}^{-1/2}\lvert\widetilde{S}_{\bm{0}}(n_{\star})\rvert\lesssim_{\epsilon}N^{(4-m)/3+\epsilon}$.

######  Remark B.2.

The $\max_{n_{\star}\mid n}$ version is not really used in the present
Appendix B (which only needs $\bm{1}_{n_{\star}=n}$), but rather in §5 (to
help separate $\bm{c}=\bm{0}$ from $\bm{c}\neq\bm{0}$).

###### Proof.

We have
$\widetilde{S}_{\bm{0}}(n_{\star})\ll_{\epsilon}n_{\star}^{1/2+\epsilon}\operatorname{cub}(n_{\star})^{m/6}$
by [hooley1988nonary]*p. 95, (170). Clearly
$\operatorname{cub}(n_{\star})\mid\operatorname{cub}(n)$ for $n_{\star}\mid
n$, so factoring out the cube-full part $n_{3}$ of $n\asymp N$, we get a bound
of

$\displaystyle\sum_{n\asymp N}n^{1-m/2+\epsilon}\operatorname{cub}(n)^{m/6}$
$\displaystyle\ll_{\epsilon}N^{\epsilon}\sideset{}{{}^{\prime}}{\sum}_{n_{3}\lesssim
N}n_{3}^{(1-m/2)+m/6}\sum_{n_{1}\asymp N/n_{3}}n_{1}^{1-m/2}$
$\displaystyle\asymp N^{\epsilon}\sideset{}{{}^{\prime}}{\sum}_{n_{3}\lesssim
N}n_{3}^{1-m/3}\cdot(N/n_{3})^{2-m/2}$ $\displaystyle\asymp
N^{\epsilon}\sideset{}{{}^{\prime}}{\sum}_{N_{3}\lesssim N}N_{3}^{1/3}\cdot
N_{3}^{m/6-1}\cdot N^{2-m/2}\asymp N^{(4-m)/3+\epsilon},$

since $1/3+(m/6-1)\geq 0$. (Here $N_{3}$ is a dyadic parameter, ranging over
powers of $2$.) ∎

######  Remark B.3.

The statement above should still hold if $\bm{0}$ is replaced with any fixed
$\bm{c}\in\mathbb{Z}^{m}$. On the other hand, $I_{\bm{0}}(n)$ behaves rather
differently than $I_{\bm{c}}(n)$ for $\bm{c}\neq\bm{0}$.

The lemma guarantees absolute convergence of the singular series

$\mathfrak{S}_{F}=\sum_{n\geq 1}n^{-m}S_{\bm{0}}(n)=\sum_{n\geq
1}n^{(1-m)/2}\widetilde{S}_{\bm{0}}(n)$

for $m\geq 5$. But as noted in [wang2021_large_sieve_diagonal_cubic_forms]*§4,
certain results from [heath1996new] imply that

$\sigma_{\infty,F,w}=\lim_{\epsilon\to 0}\,(2\epsilon)^{-1}\int_{\lvert
F(\bm{x})\rvert\leq\epsilon}d\bm{x}\,w(\bm{x})\ll_{F,w}1,$

and—if we let $\widetilde{I}_{\bm{c}}(n)\colonequals X^{-m}I_{\bm{c}}(n)$ as
in [wang2021_large_sieve_diagonal_cubic_forms]*§2.1—that
$\widetilde{I}_{\bm{0}}(n)=\sigma_{\infty,F,w}+O_{A}((n/Y)^{A})$ for all
$n\geq 1$. Here $\widetilde{I}_{\bm{0}}(n)=0$ for $n\gtrsim Y$, so

$\sum_{n\geq
1}n^{-m}S_{\bm{0}}(n)\cdot\widetilde{I}_{\bm{0}}(n)=\mathfrak{S}_{F}\cdot\sigma_{\infty,F,w}+O_{A,\epsilon}\left(\sideset{}{{}^{\prime}}{\sum}_{1\lesssim
N\lesssim
Y}N^{(4-m)/3+\epsilon}\cdot(N/Y)^{A}+\sideset{}{{}^{\prime}}{\sum}_{N\gtrsim
Y}N^{(4-m)/3+\epsilon}\right),$

with $N\in 2^{\mathbb{Z}}$. If we choose $A=(m-4)/3$, then both $N$-sums are
geometric series peaking at $N\asymp Y$, giving a final error term of
$O_{\epsilon}(Y^{(4-m)/3+\epsilon})=O_{\epsilon}(X^{(4-m)/2+\epsilon})$.

Thus (since $Y^{2}=X^{\deg F}$ and
$I_{\bm{0}}(n)=X^{m}\widetilde{I}_{\bm{0}}(n)$)

$Y^{-2}\sum_{n\geq
1}n^{-m}S_{\bm{0}}(n)I_{\bm{0}}(n)=X^{m-3}\cdot[\sigma_{\infty,F,w}\mathfrak{S}_{F}+O_{\epsilon}(X^{(4-m)/2+\epsilon})].$

If $m\geq 5$, then $(4-m)/2<0$, so we have isolated the expected singular
series from $\bm{c}=\bm{0}$.

######  Remark B.4.

One may well do better by analyzing the Dirichlet series $\sum_{n\geq
1}n^{-s}S_{\bm{0}}(n)$, using the derivative bound
$\partial_{n}^{k}I_{\bm{0}}(n)\ll_{k}n^{-k}X^{m}$ (stated in [heath1996new]*p.
183, Lemma 16 for $k=0,1$, but valid for $k\geq 0$ with little change in
proof). It would be interesting to extend the above analysis to the case $m=4$
(with powers of $\log{X}$ expected to occur for certain $F$’s).

## Acknowledgements

This paper complements Paper III; many of my acknowledgements there apply here
as well. I also thank my advisor, Peter Sarnak, for many helpful suggestions
and questions on the exposition, references, assumptions, and scope of
(various drafts of) the present work.††This work was partially supported by
NSF grant DMS-1802211.

## References