Salberger: Counting rational points on projective varieties

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corollary_0_7: On the diagonal surface a_0x_0^d+a_1x_1^d+a_2x_2^d+a_3x_3^d = 0 with nonzero rational coefficients, the points of height at most B at which no two terms a_ix_i^d and a_jx_j^d sum to zero number O_d(B^{3/sqrt d}(log B)^4+1), uniformly in the coefficients.

corollary_6_5: For nonzero rationals a_0,...,a_3, the primitive integer solutions of a_0x_0^d+a_1x_1^d+a_2x_2^d+a_3x_3^d = 0 with all |x_i| at most B and a_0x_0^d+a_jx_j^d nonzero for j = 1, 2, 3 number O_d(B^{3/sqrt d}(log B)^4+1).

theorem_0_1: For every integral projective variety X over Q of degree at least 2, the number of rational points of height at most B is O_{X,eps}(B^{dim X+eps}), the non-uniform dimension growth bound conjectured by Serre.

theorem_0_12: For a geometrically integral hypersurface X of degree d in P^{r+1} over Q and B at least 1, a single hypersurface of degree O_{d,r,eps}(B^{(r+1)/rd^{1/r}+eps}) not containing X contains every rational point of height at most B on X.

theorem_0_13: For a geometrically integral hypersurface X of degree d in P^{r+1} over Q, primes p_1,...,p_t and non-singular F_{p_i}-points P_i on the reductions of X, the points of height at most B reducing to every P_i lie on one hypersurface not containing X of degree O_{d,r}(q^{-1}B^{(r+1)/rd^{1/r}} log Bq+log Bq+1), q = p_1...p_t.

theorem_0_3: For an integral projective variety X in P^n over Q of degree d, N(X;B) is O_{d,n,eps}(B^{dim X+eps}) when d is at least 4, and O_{n,eps}(B^{dim X-1+2/sqrt 3+eps}) when d = 3.

theorem_0_4: For a polynomial f with integer coefficients in n at least 3 variables whose top-degree part is irreducible over Q of degree d, the integer zeros with all coordinates in [-B,B] number O_{d,n,eps}(B^{n-2+eps}) if d is at least 4 and O_{n,eps}(B^{n-3+2/sqrt 3+eps}) if d = 3.

theorem_0_5: On a non-singular surface of degree d in P^3 over Q, the points of height at most B off all curves of degree at most d-2 number O_d(B^{3/sqrt d}(log B)^4+1), and those off all lines number O_d(B^{3/sqrt d}(log B)^4+B).

theorem_0_6: On a non-singular complete intersection X in P^4 of hypersurfaces of degrees d_1 and d_2, with d = d_1 d_2, the points of height at most B off all curves of degree at most d_1+d_2-3 number O_{d,eps}(B^{3/sqrt d+eps}), and those off all lines number O_{d,eps}(B^{3/sqrt d+eps}+B).

theorem_0_8: On a geometrically integral projective surface of degree d in P^n over Q, the points of height at most B off all lines number O_{d,n,eps}(B^{3/sqrt d+eps}+B^{3/2sqrt d+2/3+eps}+B^{1+eps}), except for a quartic with a two-dimensional family of conics, where the bounds are O_{n,eps}(B^{43/28+eps}) and O_X(B^{3/2}).

theorem_0_9: For a geometrically integral projective surface X of degree d in P^n over Q and B at least 1, there are O_{d,n}(B^{3/2sqrt d} log B+1) geometrically integral curves of degree O_d(1) on X containing all but O_d(B^{3/sqrt d}(log B)^4+1) points of height at most B when X is a non-singular surface in P^3, and all but O_{d,n}(B^{3/sqrt d+c/log(1+log B)}) points in general.

theorem_1_2: For a geometrically integral hypersurface X of degree d in P^{r+1} over Q and a box (B_0,...,B_{r+1}), one hypersurface not containing X, of degree O_{d,r}((V/T^{1/d})^{1/rd^{1/r}} log V+1), contains every rational point of X with an integral representative in the box.

theorem_9_4: On the surface a_0x_0^d+a_1x_1^d+a_2x_2^d+a_3x_3^d = 0 over an algebraically closed field of characteristic 0 with nonzero coefficients, each locus a_0x_0^d+a_jx_j^d = 0 is a union of d^2 lines, and for d at least 3 every other closed integral curve has degree at least (d+1)/3.

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Per Salberger, "Counting rational points on projective varieties," Proceedings of the London Mathematical Society, 126(4), 1092-1133, 2023. https://doi.org/10.1112/plms.12508

Overview

Salberger studies the counting function $N(W;B)$ for rational points of standard projective height at most $B$ on quasi-projective varieties $W/\mathbb Q$. The central question is the dimension-growth conjecture: for an integral projective variety $X\subset \mathbb P^n$ of degree $d\ge 2$, should $N(X;B)$ have order at most $B^{\dim X+\varepsilon}$? The paper proves the non-uniform form

$$N(X;B)=O_{X,\varepsilon}(B^{\dim X+\varepsilon})$$

for every such $X$ (Theorem 0.1, p. 1093; its geometrically integral case is Theorem 8.13, p. 1130). The uniform conjecture is stated separately as Conjecture 0.2 (p. 1093), and is proved here for $d\ge4$:

$$N(X;B)=O_{d,n,\varepsilon}(B^{\dim X+\varepsilon}).$$

For $d=3$ the paper obtains the weaker uniform exponent $\dim X-1+2/\sqrt3+\varepsilon$ (Theorem 0.3, p. 1093; Theorem 7.5, p. 1125). Thus the fully uniform cubic case is not proved.

The principal innovation is a global determinant method using congruences simultaneously at many small primes. For a geometrically integral degree-$d$ hypersurface $X\subset\mathbb P^{m+1}$ and a coordinate box $(B_0,\ldots,B_{m+1})$, Notation 1.1 defines $V=\prod B_i$ and the largest monomial weight $T$ of a defining form (p. 1098). Theorem 1.2 (pp. 1098–1102) constructs an auxiliary hypersurface, not containing $X$ but containing all boxed rational points, of degree

$$O_{d,m}\!\left(\left(\frac{V}{T^{1/d}}\right)^{1/(m d^{1/m})}\log V+1\right).$$

Its proof compares the archimedean determinant estimate (1.12) with divisibility accumulated over all good small primes, especially (1.15) and (1.16) (pp. 1100–1101). Lemma 1.4 supplies the local $p$-adic determinant divisibility, Lemmas 1.5 and 1.9 control good reduction, and Lemma 1.11 supplies the monomial determinant estimate. The congruence-refined Theorem 2.2 (pp. 1103–1104) gains a factor $q^{-1}$ in the auxiliary degree for points specializing to prescribed nonsingular points modulo the prime factors of $q$; Lemma 2.8 (pp. 1104–1105) also controls the auxiliary hypersurface's height.

Main Lemma 3.2 (pp. 1107–1111) is the technical core. For $r\ge2$ it covers the non-singular boxed points of a hypersurface by a controlled collection of prime divisors together with the supports of effective codimension-two cycles (part (h)). Parts (a)–(h) quantify the admissible square-free moduli, the number and degrees of divisors, and the total degree of the residual cycles; part (i) gives a sharper total-degree estimate when $X$ is nonsingular. For surfaces, Theorem 3.16 (pp. 1112–1113) turns this into a covering by bounded-degree curves plus a small exceptional set. In the equal-height box, Corollaries 3.22 and 3.23 (p. 1114) give $O_{d,n}(B^{3/(2\sqrt d)}\log B+1)$ bounded-degree curves and leave $O_{d,n}(B^{3/\sqrt d+o(1)})$ points uncovered. The curve estimate used throughout is

$$N(C;B)=O_{d,n}(B^{2/d}\log B+1)$$

for an integral degree-$d$ curve, with sharper statements for $d\le2$ and for non-geometrically-integral curves (Theorem 1.17, p. 1102).

Sections 4–5 analyze the exceptional conics. Lemma 4.1 computes the hyperplane class of the $k$th Hilbert embedding of the conic Hilbert scheme (p. 1115), while Lemmas 4.2 and 4.3 constrain one- and two-dimensional families of conics (pp. 1116–1118). Lemma 5.1 bounds points on an individual conic in terms of its Hilbert height, and Lemma 5.3 sums these bounds over families (pp. 1118–1120). This geometry is combined with the surface covering theorem in Theorem 6.1 (pp. 1120–1121). In particular, for a nonsingular degree-$d$ surface $X\subset\mathbb P^3$, Theorem 6.3 and Corollary 6.4 (pp. 1121–1122) prove

$$N(U;B)=O_d(B^{3/\sqrt d}(\log B)^4+1),\qquad N(X';B)=O_d(B^{3/\sqrt d}(\log B)^4+B),$$

where $U$ omits all curves of degree at most $d-2$ and $X'$ omits all lines. The analogous complete-intersection estimates in $\mathbb P^4$ are Theorem 6.6 and Corollary 6.7 (pp. 1122–1123). The exceptional case of a quartic surface with a two-dimensional conic family receives the exponent $43/28$ in Theorem 6.1.

For affine hypersurfaces, Theorem 7.4 (pp. 1124–1125) states that if $f\in\mathbb Z[y_1,y_2,y_3]$ has highest homogeneous part irreducible over $\overline{\mathbb Q}$ of degree $d$, then the number of zeros in $[-B,B]^3$ is $O_{d,\varepsilon}(B^{1+\varepsilon})$ for $d\ge4$, and $O_\varepsilon(B^{2/\sqrt3+\varepsilon})$ for $d=3$. Hyperplane-section and birational-projection arguments then yield the uniform projective bounds of Theorem 7.5.

The remaining non-uniform cubic case is handled by methods distinct from the determinant argument. Section 8 analyzes cubic forms containing rational lines, reducing suitable affine counts to families of binary quadratics or to divisor estimates; see Theorem 8.11 (p. 1129). Together with cited results for the complementary cases, this gives

$$n(G;B)=O_{G,\varepsilon}(B^{n-2+\varepsilon})$$

for every absolutely irreducible cubic form $G$ (Theorem 8.12, p. 1129), and hence Theorem 8.13. These cubic conclusions are non-uniform in the form.

Finally, Section 9 is an independent geometric application to diagonal surfaces. Theorem 9.4 (p. 1131) identifies the $3d^2$ standard lines on a nonsingular diagonal degree-$d$ surface and proves, for $d\ge3$, that every other integral curve has degree at least $(d+1)/3$. Combined with the surface estimates, this yields the coefficient-uniform diagonal bound in Corollary 6.5 (p. 1122). Results attributed to earlier papers—such as the local determinant bounds recalled as Theorems 0.10 and 0.11—are background rather than new theorems of this paper.

Relation to E940

This source bears on Problem 940.

Salberger's bounds count points of bounded height on varieties; they do not bound the number of integers up to $X$ that are sums of at most $r$ $r$-powerful numbers, which is what E940 asks about.

Bears on. #940: the paper's bounds count points of bounded height on varieties; none bounds the number of integers represented as sums of $r$-powerful numbers, and the paper settles neither question of the problem.

Results. Labels and pages are those of the journal print (pp. 1092–1133). Each page below gives the statement, a proof pointer and its read depth (claims checked).

• Theorem 0.1 (p. 1093): $N(X;B)=O_{X,\varepsilon}(B^{\dim X+\varepsilon})$ for every integral projective variety of degree $d\ge2$ over $\mathbb Q$.
• Theorem 0.3 (p. 1093; Theorem 7.5, p. 1125): the uniform bounds for $d\ge4$ and $d=3$, with Conjecture 0.2 (p. 1093).
• Theorem 0.4 (p. 1093; Theorem 7.4, pp. 1124–1125): integer zeros in $[-B,B]^n$ of a polynomial with irreducible top-degree part.
• Theorem 0.5 (p. 1094; Theorem 6.3 and Corollary 6.4, pp. 1121–1122): non-singular surfaces in $\mathbb P^3$.
• Theorem 0.6 (p. 1094; Theorem 6.6 and Corollary 6.7, pp. 1122–1123): non-singular complete intersection surfaces in $\mathbb P^4$.
• Corollary 0.7 (pp. 1094–1095): diagonal surfaces, uniformly in the coefficients.
• Theorem 0.8 (p. 1095; Theorem 6.1, pp. 1120–1121): geometrically integral surfaces in $\mathbb P^n$, with the quartic exception.
• Theorem 0.9 (p. 1095): covering by $O_{d,n}(B^{3/2\sqrt d}\log B+1)$ curves of bounded degree.
• Theorem 0.12 (p. 1096): one auxiliary hypersurface for all points of height at most $B$.
• Theorem 0.13 (p. 1096): the same with prescribed non-singular reductions.
• Theorem 1.2 (p. 1098): the box version of the global determinant method.
• Corollary 6.5 (p. 1122): primitive solutions of diagonal quaternary equations.
• Theorem 9.4 (p. 1131): degrees of curves on diagonal surfaces.

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