The Halphen cubics of order two

Abstract

For each $m\ge 1$, Roulleau and Urzúa give an implicit construction of a configuration of $4(3m^2-1)$ complex plane cubic curves. This construction was crucial for their work on surfaces of general type. We make this construction explicit by proving that the Roulleau-Urzúa configuration consists precisely of the Halphen cubics of order $m$, and we determine specific equations of the cubics for $m=1$ (which were known) and for $m=2$ (which are new).

20 pages