[論文レビュー] Strong monodromy conjecture for defining polynomials of projective hypersurfaces having only weighted homogeneous isolated singularities

Morihiko Saito|arXiv (Cornell University)|Feb 27, 2026

Algebraic Geometry and Number Theory被引用数 0

ひとこと要約

The paper proves the strong monodromy conjecture for defining polynomials of reduced projective curves with weighted homogeneous isolated singularities by combining Denef–Loeser’s Newton-nondegenerate three-variable formula with two-variable results, observing an unexpected cancellation that prevents counterexamples.

ABSTRACT

Let $Z\subset{\bf P}^{n-1}$ be a hypersurface such that the associated reduced hypersurface $Z_{ m red}$ has only weighted homogeneous isolated singularities. In the case $Z$ is a reduced curve or $Z_{ m red}$ has only homogeneous isolated singularities with $n$ at least $4$, we show that the strong monodromy conjecture for a defining polynomial $f$ of $Z$ follows immediately from arxiv:1609.04801v1 using in the reduced curve case a formula of Denef and Loeser for Newton-nondegenerate polynomials of three variables (which can be deduced in the applied case from the one for the two variable case) together with known results about the strong monodromy conjecture in the two variable case. Here an amazing cancellation occurs so that possible counterexamples fail.

研究の動機と目的

• Motivate the study by Bernstein–Sato polynomials and topological zeta functions for projective curves with weighted homogeneous singularities.
• Investigate when the strong monodromy conjecture holds for defining polynomials of reduced projective curves in P^2.
• Leverage Newton-nondegenerate techniques and known two-variable results to derive global conclusions from local data.
• Show cancellations in the three-variable Newton-nondegenerate case that prevent potential counterexamples.

提案手法

• Use embedded resolution theory to define the local topological zeta function Z_top,f,0(s).
• Apply Denef–Loeser’s formula for Newton-nondegenerate polynomials in three variables to express Z_top,f,0(s) via faces of the Newton polytope.
• Relate the three-variable formula to the two-variable case by blowups and line-center resolutions, enabling transfer of known results.
• Compute explicit J_sigma(s) terms for the relevant compact faces of the Newton polytope in the three-variable setting.
• Demonstrate that potential poles cancel or become poles of local zeta functions at singular points, ensuring alignment with Bernstein–Sato roots.
• Discuss reductions to homogeneous cases and implications for non-reduced curves via auxiliary lemmas.

実験結果

リサーチクエスチョン

• RQ1 Does the strong monodromy conjecture hold for defining polynomials of reduced projective curves in P^2 with only weighted homogeneous singularities?
• RQ2 Can Denef–Loeser’s Newton-nondegenerate three-variable framework, together with two-variable results, force the poles of Z_top,f,0(s) to be roots of b_f(s)?
• RQ3 How do cancellations in the Newton polytope analysis prevent potential counterexamples in the three-variable case?
• RQ4 What reductions to homogeneous or cone-like structures simplify the verification of the conjecture for these curves?

主な発見

• The strong monodromy conjecture for f follows from the three-variable Newton-nondegenerate formula together with two-variable results.
• An amazing cancellation occurs so that potential counterexamples to the conjecture do not occur.
• In the applied (three-variable) case, a pole at -3/d can disappear in Z_top,f,0(s) despite nonvanishing Euler number, due to the cancellation behavior.
• Theorem 3 asserts that any pole of Z_top,f,0(s) is a pole of the local topological zeta function at a singular point of C.
• The approach links the three-variable Newton-nondegenerate case to the two-variable case via blowups and line-center resolutions, with computations feasible in computer algebra systems.
• Proposition 1 states a broader validity: for reduced Z with homogeneous isolated singularities (not ordinary double points) in dimension n ≥ 4, the strong monodromy conjecture holds for f.