# Slope equality of plane curve fibrations and its application to Durfee's conjecture

## TL;DR

This paper establishes a slope equality for fibered surfaces with smooth plane curve fibers and applies it to prove a strong Durfee-type inequality for certain surface singularities, advancing understanding in algebraic geometry.

## Contribution

It introduces a new slope equality for fibered surfaces with plane curve fibers and applies it to prove a strong Durfee's conjecture for specific hypersurface singularities.

## Key findings

- Proved a slope equality for fibered surfaces with smooth plane curve fibers.
- Established a strong Durfee-type inequality for isolated hypersurface surface singularities.
- Confirmed Durfee's strong conjecture for singularities with non-negative Euler number of the exceptional set.

## Abstract

We give a slope equality for fibered surfaces whose general fiber is a smooth plane curve. As a corollary, we prove a "strong" Durfee-type inequality for isolated hypersurface surface singularities, which implies Durfee's strong conjecture for such singularities with non-negative topological Euler number of the exceptional set of the minimal resolution.