Algebraic k-systems of curves

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 $g^2$.