Non-general type surfaces in P4, an update

Hirotachi Abo; Kristian Ranestad; Frank-Olaf Schreyer

Non-general type surfaces in P4, an update

Algebraic Geometry 2026-07-18 v1

Authors: Hirotachi Abo , Kristian Ranestad , Frank-Olaf Schreyer

Abstract

A general algebraic surface cannot be embedded in P4. Proving a conjecture by Hartshorne and Lichtenbaum, Ellingsrud and Peskine showed that there is a degree bound for smooth rational surfaces in P4, and in fact for surfaces not of general type. We give a survey of the classification status and of classical and computer-aided constructions of smooth non-general type surfaces in P4.

Cite

@article{arxiv.2607.16753,
  title  = {Non-general type surfaces in P4, an update},
  author = {Hirotachi Abo and Kristian Ranestad and Frank-Olaf Schreyer},
  journal= {arXiv preprint arXiv:2607.16753},
  year   = {2026}
}

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35 pages