An arXiv preprint proves a family of exact blowup equations for local Hirzebruch threefolds. The result covers the inclusive range of Hirzebruch indices from 0 through 2. At index 2, the statement is made through localized indices because the paper establishes properness for the fixed locus but not for the full stable-pair moduli space.
The paper compares two kinds of mathematical object: stable-pair vertex sums and the moduli space M(2,n) of framed rank-two sheaves on P2, the projective plane. The comparison links the stable-pair series to the framed-sheaf series used in the blowup argument.
A coefficient-by-coefficient match
The proof uses a four-chart preferred-slope localization, divides by the contribution of the fibre class, and then invokes K-theoretic blowup formulas for framed sheaves. This framework turns the comparison into a calculation involving matching formal series coefficient by coefficient.
After fibre and degree-zero normalization, the stable-pair series agrees with the framed-sheaf series coefficient by coefficient in the base variable across the same Hirzebruch range. At index 2, that equality is an equality of localized indices, reflecting the narrower properness result.
That comparison leads to bilinear blowup identities for every allowed shift in the Hirzebruch cases. The shift labelled (1, 1) gives a vanishing equation for class (2, 0): the corresponding coefficient is exactly zero. These are coefficientwise mathematical identities, and an analytic reading would require compatible meromorphic continuation and normal convergence.
Local P2 remains conditional
Local P2 is more tentative. Under Conjectures 1 and 2, the paper obtains three equations indexed by r=1, -1 and 3. Two have nonzero unity factors, displayed as exponentials with exponents s/18 and -s/18; the factor for r=3 is zero. Here s is the sum of the two additive equivariant parameters.
Those conditional equations also produce finite coefficient recursions. Together with the degree-zero starting value A0=1, they uniquely determine the local-P2 coefficient series, because the determinant displayed in the paper is nonzero in the relevant coefficient field.
The first condition is substantial: Conjecture 1 is not proved because the required relative K-theoretic stable-pair degeneration formula, with compatible square-root control, is unavailable in the needed generality. Conjecture 2 is also unproved. Its positive-exponent product representations and compatibility with the entire bilateral sum remain assumptions.
What the paper leaves open
The paper reports explicit formulas for A1 and A2 and says that four transformed Laurent-polynomial entries agree with cited tables. It also notes that a full geometric identification of the relevant index would require further comparison of conventions.
Two broader directions remain conjectural. One is a general Huang-Sun-Wang blowup statement for a local Calabi-Yau threefold. The other concerns the local rational elliptic surface and uses the E8 lattice, its 240 roots, the zero vector and theta-function coefficients. These are proposed extensions, not results established by the paper.
The supplied document is an arXiv preprint, version 2, dated 10 September 2026. The author acknowledges support from JSPS KAKENHI Grant Number JP25K17226. Within the stated scope, the proved result is the coefficientwise stable-pair blowup theory for the Hirzebruch geometries, while local P2 and the broader conjectures still depend on the assumptions or further proofs identified in the paper.
Paper data and sources
Original title: Topological String Blowup Equations via Stable Pairs
Authors: Lutian Zhao
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text