Algebraic k-systems of curves

TL;DR

This paper determines the maximum size of algebraic k-systems of curves on surfaces, revealing exact bounds depending on genus and parity of k, and provides constructions illustrating the bounds' tightness.

Contribution

It generalizes a previous theorem to compute maximum sizes of algebraic k-systems on surfaces, depending on genus and parity of k.

Findings

Maximum size is 2g+1 for g≥3 or odd k

Maximum size is 2g otherwise

Constructs of curves with pairwise geometric intersections of two grow as g^2

Abstract

A collection $ \Delta $ of simple closed curves on an orientable surface is an algebraic $ k $-system if the algebraic intersection number $\langle \alpha,\beta \rangle$ is equal to $k $ in absolute value for every $ \alpha , \beta \in \Delta $ distinct. Generalizing a theorem of [MRT14] we compute that the maximum size of an algebraic $k$-system of curves on a surface of genus $g$ is $2g+1$ when $g\ge 3$ or $k$ is odd, and $2g$ otherwise. To illustrate the tightness in our assumptions, we present a construction of curves pairwise geometrically intersecting twice whose size grows as $g^2$.