Algebraic intersection for hyperbolic surfaces

TL;DR

This paper investigates the behavior of the algebraic intersection form on hyperbolic surfaces, revealing its minimal value in moduli space and its asymptotic growth as certain geometric parameters tend to zero.

Contribution

It establishes the existence of a minimum for the algebraic intersection form in moduli space and characterizes its asymptotic growth with respect to genus and systolic length.

Findings

Minimum of intersection form grows as (log g)^{-2} with genus g

Asymptotic behavior described as homologically systolic length approaches zero

Provides new insights into geometric properties of hyperbolic surfaces

Abstract

We show that the algebraic intersection form of hyperbolic surfaces of genus $g$ has a minimum in the moduli space and that the minimum grows in the order $(\log g)^{-2}$ in terms of the genus. We also describe the asymptotic behavior of the algebraic intersection form in the moduli space as the homologically systolic length goes to zero.