The Halphen cubics of order two

TL;DR

This paper explicitly identifies the Roulleau-Urzúa configuration of complex plane cubics as Halphen cubics of order m, providing explicit equations for m=1 and m=2, advancing understanding of these configurations in algebraic geometry.

Contribution

The paper proves that the Roulleau-Urzúa configuration exactly matches the Halphen cubics of order m and derives explicit equations for these cubics for m=1 and m=2.

Findings

Confirmed the Roulleau-Urzúa configuration as Halphen cubics of order m

Derived explicit equations for m=1 and m=2 cubics

Enhanced understanding of cubic configurations in algebraic geometry

Abstract

For each $m\ge 1$, Roulleau and Urz\'ua 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-Urz\'ua 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).