General Castelnuovo bound for polarized smooth projective threefolds

Prove that for every polarized smooth projective complex threefold (X,H), when H generates NS(X)_{mathrm{tf}} and s is the least positive integer for which a nonzero effective divisor is numerically equivalent to sH, every one-dimensional closed subscheme C of degree d and genus g satisfies g <= d^2/(2sH^3) + (s + K_X.H^2/H^3)d/2 + 1 without any additional constant term.

Background

The paper states a general Castelnuovo Bound Conjecture for polarized smooth projective complex threefolds. It incorporates the least numerical effective-divisor multiple s and predicts both the optimal quadratic and linear coefficients in the genus bound.

Theorem~ref{thm:cast-bound} establishes the same leading terms only after adding a computable constant depending on the Bogomolov–Gieseker inequality constant. Thus, the exact conjectural inequality, with constant term 1 and no additional error term, is not proved in the paper.

References

Finally, we discuss consequences of Theorem~\ref{thm:main1.1} concerning Castelnuovo Bound Conjecture. One of its general form can be stated as follows.

Let (X,H) be a polarized smooth projective complex threefold. Assume that H generates NS(X)\mathrm{tf} and s\in Z{\geq 1} is the least integer such that there exists a nonzero effective divisor numerically equivalent to sH. Then for any one-dimensional closed subscheme C\subset X of degree d and genus g, we have g\leq \frac{1}{2sH3}d2+\frac{s+\frac{K_X.H2}{H3}}{2}d+1.

Using Theorem~\ref{thm:main1.1}, we can prove Conjecture~\ref{conj:general-cast} up to an computable constant.

— Generalized Bogomolov--Gieseker inequalities via Frobenius  (2610.18595 - Li et al., 13 Oct 2026) in Section 6, subsection “Castelnuovo Bound Conjecture”; Conjecture~ref{conj:general-cast}