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arXiv · 1603.04480

The Halphen cubics of order two

Thomas Bauer·Brian Harbourne·Joaquim Roé·Tomasz Szemberg
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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).

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Thomas Bauer, Brian Harbourne, Joaquim Roé, Tomasz Szemberg. 2016-03-14. The Halphen cubics of order two. https://arxiv.org/abs/1603.04480

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