A mathematical construction has opened a broad range of possible Chern slopes, the ratio c₁²/c₂ attached to a surface, for complex surfaces whose fundamental group is prescribed in advance. The paper proves that, for each such group G, those slopes are dense across [1/2, 3]. Dense means values can be found arbitrarily close to any target in that interval, rather than only at a few isolated points.
The result is more limited when the canonical class must be ample, the paper’s term for an added positivity condition. In that case, the proved dense range is [1/2, 2].
A geometric recipe
The objects are mathematical families, not an empirical sample: minimal, nonsingular projective surfaces of general type S with π₁(S) ≃ G. The central question is which Chern slopes can occur while that topological fundamental group is fixed in advance, with special attention to ample canonical classes.
At the center is a product construction. It takes a lef line bundle σ of exponent 1 on X and a very ample line bundle B on Y, combines their pullbacks as M = pr₁*σ ⊗ pr₂*B, and intersects two general members of |M| to form S. The resulting surface has π₁(S) ≃ π₁(X) × π₁(Y), and its canonical class K_S is big and nef.
Ampleness gets an exact test. K_S is ample if and only if X has no irreducible curve Γ for which both K_X·Γ and σ·Γ are zero. That turns the construction’s positivity requirement into a concrete obstruction to check.
The input X is supplied by a cover branched over a smooth curve, with ample K_X and no (-2)-curves; the simplest example is a double cover of P¹ × P¹. In the stated family, X is nonsingular with trivial fundamental group, while σ is ample, globally generated and lef, with exp(σ) = 1.
How the interval emerges
For parameters u, v, a and b, the input family has c₁²(X) = 4uv, c₂(X) = 8uv + 12u + 12v + 24, σ² = 4ab and σ·K_X = 2(av + bu). These formulas provide the numerical inputs for the later slope calculation.
As the construction parameter m tends to infinity, the ratio c₁²(S)/c₂(S) approaches a limit Λ(X,σ). The numerator receives 24σ² + 12σ·K_X, while the denominator receives 18σ² + 6σ·K_X. The limiting value is independent of Y and B₀, leaving the chosen X and σ to determine it.
The final parameter calculation reduces to μ(y) = (1 + 6y)/(2 + 3y). The function is strictly increasing and maps positive y onto the open interval (1/2, 2). That monotonic sweep underpins density in [1/2, 2] for the ample-canonical case, allowing interior target slopes to be approached by tuning y.
The gap above 2
The ample-canonical result does not close the full range. Density on [2, 3] remains open and, within this method, is reduced to Question 6.1: whether the surfaces X_p carry very ample line bundles σ_p satisfying the two stated small-order conditions.
At the lower end, the authors place the result alongside a cited Mendes Lopes–Pardini result and Reid’s conjecture. They say [1/3, 3] is the largest conceivable interval in the arbitrary-G discussion, while density below 1/2 would conflict with Reid’s conjecture for the chosen G. The result therefore leaves density below 1/2 tied to that unresolved context.
One earlier construction is also ruled out as an ample route: the surfaces S_p from the cited construction never have ample canonical class, regardless of their defining sections. The supplied document is an arXiv preprint, version 1, dated 26 August 2026.
Paper data and sources
Original title: The geography of Chern slopes with prescribed fundamental group
Authors: Maycol Falla Luza
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text