Algebraic k-systems of curves

A collection $Δ$ of simple closed curves on an orientable surface is an algebraic $k$-system if the algebraic intersection number $\langle α,β\rangle$ is equal to $k$ in absolute value for every $α, β\in Δ$ 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 $g^2$.

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