A mathematical preprint reports a universal way to organize tautological pushforwards from relative Hilbert schemes in families of curves and surfaces. A pushforward here means carrying the result of a calculation on the Hilbert scheme back to the base of the family. The paper packages those results in universal combinations of relative κ-classes. For surfaces, its main generating series is the exponential of such a κ-class combination, with coefficients that are uniquely determined.
One recipe for many families
The proof combines relative incidence reduction, disjoint-union additivity and test families to establish universality and uniqueness. It is a geometric and algebraic route to the result, built around identities for structured mathematical families.
The surface theorem is stated over every smooth, connected, quasi-projective base, every smooth projective surface family and every K-theory class of fixed rank. The universal series is therefore asserted across all of those qualifying inputs rather than for one selected surface or one special class.
Curves make the recursion visible
On curves, the paper gives a logarithmic version of the same idea. Taking the logarithm of the total Segre pushforward produces a universal linear expansion in relative κ-classes. The coefficients are indexed by powers of the line-bundle class and the canonical class, organizing a family of related terms instead of listing isolated formulas.
The first curve coefficients are fixed after introducing the coordinate q = u(1 - u). In that coordinate, A_{1,0} is log(1 - u), while A_{0,1} combines a negative log(1 - u) with a half-log term involving 1 - 2u. For b at least 2, a Δ-recursion determines the higher edge coefficients, with A_{0,b}(0) = 0 as the normalization. A second, horizontal relation carries the calculation from lower to higher line-bundle degree: when a is at least 1 and a + b is at least 2, A_{a,b} is obtained by applying negative one over a times (2θ + a + b - 2) to A_{a-1,b}.
That edge recursion also has a structural interpretation. The paper identifies it as the geometric counterpart of the principal specialization of the monotone Hurwitz quantum curve, linking the curve-family calculation to that quantum-curve structure.
The surface calculation
Surfaces add a second layer of coefficient bookkeeping. Their horizontal relation moves in the line-bundle direction: for a at least 1 and a + b + 2c at least 3, A_{a,b,c} is obtained by applying negative one over a times (3θ + a + b + 2c - 3) to A_{a-1,b,c}. The recursion yields explicit expressions in v for the normalized codimension-one coefficients with positive h-degree.
The two remaining normalized codimension-one coefficients are also given explicitly, as rational functions of v. To compute them, the paper combines torus localization of an affine-plane vertex with refined topological recursion, while the horizontal recursion supplies the separation relation. The authors characterize these codimension-one formulas as a complete solution for arbitrary smooth projective surface families.
The explicit surface formulas concern codimension one, while the horizontal relation is the stated mechanism for moving between line-bundle degrees. The document also records further questions and says that higher-rank and multiple-insertion variants were not submitted to current models.
A preprint with explicit boundaries
The work is an arXiv version v1 preprint dated 26 Aug 2026. Its AI disclosure attributes the mathematical content to human work. It says GPT-5.4 and GPT-5.5 were used for literature searches and explanations, and GPT-5.6 and Codex for drafting and related tasks, with the author taking responsibility.
Paper data and sources
Original title: Tautological Pushforwards of Hilbert Schemes of Points on Curves and Surfaces
Authors: Bochao Kong
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text