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Self-intersection Number of Negative Curves on Fermat Surfaces

Zhenjian Wang

TL;DR

The paper provides an explicit formula for the self-intersection number $C^2$ of negative curves on Fermat surfaces $X_d$ in terms of a unique plane curve $E$ of degree $e$, Milnor numbers, and intersection data with the Fermat curve $D$. The approach reduces the problem to plane-curve data via a degree $d$ Galois cover $\rho:X_d\to\mathbb{P}^2$, analyzes the pullback divisor $\mathscr{C}=\rho^*E$, and decomposes it into $k$ Galois-conjugate components, all sharing the same self-intersection. A local Puiseux-series and resultant/discriminant framework is developed to compute the intersection multiplicities of these components, yielding an explicit global formula for $C^2$; in particular, when $d$ is prime one obtains a simplified expression. The results provide sufficient conditions for the Bounded Negativity Conjecture on Fermat surfaces and introduce an invariant $I_f$ to quantify singularity contributions, linking negativity to plane-curve singularity theory with potential implications for counterexamples or bounds in higher-degree surfaces.

Abstract

We give an explicit formula for the self-intersection number of negative curves on Fermat surfaces. The formula offers us hints to either prove or disprove the Bounded Negativity Conjecture for the Fermat surfaces.

Self-intersection Number of Negative Curves on Fermat Surfaces

TL;DR

The paper provides an explicit formula for the self-intersection number of negative curves on Fermat surfaces in terms of a unique plane curve of degree , Milnor numbers, and intersection data with the Fermat curve . The approach reduces the problem to plane-curve data via a degree Galois cover , analyzes the pullback divisor , and decomposes it into Galois-conjugate components, all sharing the same self-intersection. A local Puiseux-series and resultant/discriminant framework is developed to compute the intersection multiplicities of these components, yielding an explicit global formula for ; in particular, when is prime one obtains a simplified expression. The results provide sufficient conditions for the Bounded Negativity Conjecture on Fermat surfaces and introduce an invariant to quantify singularity contributions, linking negativity to plane-curve singularity theory with potential implications for counterexamples or bounds in higher-degree surfaces.

Abstract

We give an explicit formula for the self-intersection number of negative curves on Fermat surfaces. The formula offers us hints to either prove or disprove the Bounded Negativity Conjecture for the Fermat surfaces.
Paper Structure (13 sections, 9 theorems, 87 equations)

This paper contains 13 sections, 9 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Curves on surfaces of Sebastiani-Thom type
  3. Strategy of Proof of Theorem \ref{['BNCFermat']}
  4. Strategy
  5. Proof of Lemma \ref{['w0']}
  6. Self-intersection number for curves on Fermat surfaces
  7. Position of intersection points
  8. Intersection multiplicities
  9. An invariant
  10. Puiseux series, resultants, and intersection multiplicities
  11. Puiseux series
  12. Resultants, discriminants, and intersection multiplicities
  13. Proof of Proposition \ref{['intnum']}

Key Result

Theorem 1.1

Let $d\geq5$. Suppose that $C$ is a reduced and irreducible curve on the Fermat surface $X_d$ with self-intersection number $C^2<2-d<0$. Then the following properties hold.

Theorems & Definitions (18)

  • Theorem 1.1: Theorem \ref{['main']}
  • Conjecture 1.2: Bounded Negativity Conjecture
  • Conjecture 1.3: Bounded Negativity Conjecture for integral curves
  • Corollary 1.4: Corollary \ref{['sufcond']}
  • Conjecture 1.5
  • Conjecture 1.6
  • Proposition 2.1
  • proof
  • Lemma 3.2
  • Lemma 4.2
  • ...and 8 more