Beyond the square-root barrier: cubic forms of Perazzo type

Abstract: We show how the circle method can be used to study rational points on a certain cubic fourfold, going beyond the square-root barrier.

• The paper establishes an unconditional asymptotic for rational points on a singular Perazzo-type cubic fourfold, pushing analysis past the square-root barrier.
• It innovatively applies a smooth δ-method variant of the circle method to achieve strong cancellation in finite field exponential sums.
• The results confirm the Manin conjecture’s predictions by linking geometric singularities and dual variety contributions with refined analytic estimates.

Cubic Forms of Perazzo Type Beyond the Square-root Barrier

Introduction and Context

The study of rational points on cubic hypersurfaces has been a central topic in arithmetic geometry and analytic number theory. Quantitative results for counting rational points of bounded height on these varieties are fundamentally constrained by analytic techniques, particularly the so-called "square-root barrier" encountered in circle method arguments. This essay examines the results and methods of "Beyond the square-root barrier: cubic forms of Perazzo type" (2604.19045), which provides an unconditional estimate for the number of rational points of bounded height on a singular cubic fourfold of the so-called Perazzo type, achieving results beyond the square-root barrier.

The central object of study is the cubic fourfold $X \subset \mathbb{P}^5$ defined over $\mathbb{Q}$ by the equation

$F(x, y) = x_1y_1^2 + x_2y_2^2 + x_3y_3^2 = 0$

where $x, y \in \mathbb{Z}^3$, a non-conical cubic with a singular locus. The paper focuses on the asymptotic behaviour of the rational point counting function $N(B)$ with respect to a smooth, compactly supported weight function $W$: $$N(B) = \sum_{\substack{x,y \in \mathbb{Z}^3 \ F(x, y) = 0}} W\left(\frac{(x, y)}{B}\right).$$ The main theorem establishes an asymptotic with a logarithmic main term, in full accordance with the Manin conjecture's predictions for the canonical crepant resolution of $X$.

Circle Method Beyond the Square-root Barrier

The classical Hardy-Littlewood circle method encounters substantial difficulties when the number of variables is small. For cubic hypersurfaces in $\mathbb{P}^{n-1}$, the method often cannot penetrate the square-root cancellation threshold unless $n > 6$. However, the cubic form $\mathbb{Q}$0 under study, as anticipated by a suite of algebraic and analytic arguments, exhibits exponential sums with better-than-square-root cancellation.

A key innovation is the application of the smooth $\mathbb{Q}$1-method variant of the circle method—specifically, Heath-Brown's form based on the $\mathbb{Q}$2-symbol as in [HB]. The method introduces an auxiliary modulus $\mathbb{Q}$3 and a smoothing function $\mathbb{Q}$4, followed by Poisson summation over the lattice variables, leading to the analysis of exponential sums of the form: $\mathbb{Q}$5 and corresponding analysis of oscillatory integrals $\mathbb{Q}$6.

The essential technical advance is the proof that, for this particular family, the exponential sums $\mathbb{Q}$7 satisfy uniform bounds showing complete cancellation (i.e., $\mathbb{Q}$8 rather than $\mathbb{Q}$9). This phenomenon is formalized as property $F(x, y) = x_1y_1^2 + x_2y_2^2 + x_3y_3^2 = 0$0, which is closely related to the vanishing of the $F(x, y) = x_1y_1^2 + x_2y_2^2 + x_3y_3^2 = 0$1-number in the sense of Katz [katz], and its verification for Perazzo-type cubic forms is nontrivial and central to the argument.

Main Asymptotic and the Manin Conjecture

The principal result is the following unconditional asymptotic, as $F(x, y) = x_1y_1^2 + x_2y_2^2 + x_3y_3^2 = 0$2: $F(x, y) = x_1y_1^2 + x_2y_2^2 + x_3y_3^2 = 0$3 for any smooth weight $F(x, y) = x_1y_1^2 + x_2y_2^2 + x_3y_3^2 = 0$4, where $F(x, y) = x_1y_1^2 + x_2y_2^2 + x_3y_3^2 = 0$5 is the weighted real density of solutions in $F(x, y) = x_1y_1^2 + x_2y_2^2 + x_3y_3^2 = 0$6 and $F(x, y) = x_1y_1^2 + x_2y_2^2 + x_3y_3^2 = 0$7 denotes the Riemann zeta function.

An extensive analysis confirms that this result matches the predictions of the Manin conjecture for a crepant resolution $F(x, y) = x_1y_1^2 + x_2y_2^2 + x_3y_3^2 = 0$8, incorporating Peyre's conjectural refinements for the leading constant (see [manin, peyre]). A meticulous computation of Tamagawa factors, effective cone volumes, and component measures for $F(x, y) = x_1y_1^2 + x_2y_2^2 + x_3y_3^2 = 0$9 yields complete agreement between the geometric/arithmetic conjecture and the analytic computation.

A particularly important phenomenon is that only $x, y \in \mathbb{Z}^3$0 of the main term arises from the contribution with $x, y \in \mathbb{Z}^3$1 in the dual Poisson decomposition, with the remaining $x, y \in \mathbb{Z}^3$2 captured from nontrivial dual vectors satisfying $x, y \in \mathbb{Z}^3$3, where $x, y \in \mathbb{Z}^3$4 is a specific sextic dual form defining the dual variety of $x, y \in \mathbb{Z}^3$5. This split is mathematically grounded in the geometric properties of the hypersurface and appears analogously in related settings treated via the circle method.

Finite Field Exponential Sums and Property $x, y \in \mathbb{Z}^3$6

A central technical component involves the explicit analysis and bounding of exponential sums associated to $x, y \in \mathbb{Z}^3$7 over finite fields. For the major arcs, one requires effective cancellation in $x, y \in \mathbb{Z}^3$8 for all but a thin subset of parameters. The paper identifies several families of cubic forms in dimension 6 that possess property $x, y \in \mathbb{Z}^3$9, including:

• Cubic fourfolds singular along a rational plane,
• Families related to pencils of quadric surfaces,
• Certain trilinear forms,
• Forms of the archetype $N(B)$0, with the most generic cases especially intricate due to limited linear structure.

The explicit analysis of the exponential sums leverages character sum evaluations, large sieve arguments, and the arithmetic of the duality between $N(B)$1 and its dual variety $N(B)$2.

Technical Devices: Delta-Function Sieve and Hooley's $N(B)$3-Function

To treat error terms and secondary main contributions, the paper employs Hooley's $N(B)$4-function and related machinery from divisor function theory. The key challenge is to avoid incurring extra logarithmic losses that typically arise in divisor sum estimates along polynomial sequences—a difficulty resolved by invoking uniformity results in recent work of de la Bretèche--Tenenbaum [regis], among others.

The auxiliary estimates for the number of zeros of polynomials modulo prime powers, and bounds for character sums over variable moduli, are essential to fully exploit the structure of the cubic form and complete the analytic estimates required for unconditional main term extraction.

Comparison with Prior Work and Broader Implications

The methods and explicit main term established for this Perazzo-type cubic fourfold provide the first unconditional circle method proof for a reduced, irreducible cubic fourfold in $N(B)$5 variables. Prior arithmetic investigations of related singular cubic fourfolds (e.g., [BBS], [ulrich]) have established Manin-type asymptotics for certain highly symmetric or toric examples, albeit with different main term orders and via alternative methods (lattice point counts, Mellin analysis, universal torsors). However, the approach here is both more general and analytically deeper: it leverages algebraic cancellation in the finite field exponential sums to push the circle method into previously inaccessible territory.

The dual variety's role in the arithmetic decomposition of rational points is elucidated through the detailed analysis of minor arc and nontrivial dual vector contributions. This aspect has deep connections to the distribution of rational planes and lines in higher-degree hypersurfaces, and it illuminates the subtle interplay between geometric singularities and analytic behavior in the context of rational point counting.

From a theoretical perspective, this work isolates a class of cubic fourfolds whose arithmetic is amenable to a direct circle method attack, suggesting the existence of a relatively explicit set of conditions (related to property $N(B)$6) under which the square-root barrier can be universally breached for cubic forms in six variables. The identification of such forms—both in dimension and in terms of their singular loci—suggests new directions for the classification of Diophantine problems tractable by analytic methods.

Conclusion

"Beyond the square-root barrier: cubic forms of Perazzo type" (2604.19045) provides an unconditional, analytic proof of the Manin conjecture for a singular cubic fourfold in six variables, fully reconciling the arithmetic, geometric, and analytic predictions for such hypersurfaces. The paper's technical advances—the verification of strong cancellation in finite field exponential sums, the navigation of dual variety contributions, and the refined use of the smooth circle method—represent significant progress in understanding the reach of analytic number theory for singular and low-dimensional higher-degree varieties. The identification and analysis of the special cancellation property $N(B)$7 define a promising path for future work targeting cubic forms and beyond, potentially guiding developments on rational points in even more complex geometric settings.