The Halphen cubics of order two

Thomas Bauer, Brian Harbourne, Joaquim Roé, Tomasz Szemberg

Research output: Contribution to journalArticleResearchpeer-review


© 2016, Universitat de Barcelona. For each m⩾ 1 , Roulleau and Urzúa give an implicit construction of a configuration of 4 (3 m2- 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).
Original languageEnglish
Pages (from-to)339-357
JournalCollectanea Mathematica
Publication statusPublished - 1 Sep 2017


  • Abelian surfaces
  • Automorphisms
  • Halphen cubics
  • Hesse arrangement
  • Rational surfaces


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