Convex hulls of surfaces in fourspace

• Meroni, Chiara ^[3] ; Ranestad, Kristian ^[1] ; Sinn, Rainer ^[2]
  1. [1] University of Oslo

     University of Oslo

     Noruega

     
  2. [2] University of Leipzig

     University of Leipzig

     Kreisfreie Stadt Leipzig, Alemania

     
  3. [3] ETH Institute for Theoretical Studies, Zürich, Switzerland

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• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 76, Fasc. 3, 2025, págs. 545-574
• Idioma: inglés
• DOI: 10.1007/s13348-024-00444-w
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• Resumen

  • This is a case study of the algebraic boundary of convex hulls of varieties. We focus on surfaces in fourspace to showcase new geometric phenomena that neither curves nor hypersurfaces exhibit. Our method is a detailed analysis of a general purpose formula by Ranestad and Sturmfels in the case of smooth real algebraic surfaces of low degree (that are rational over the complex numbers). We study both the complex and the real features of the algebraic boundary of Veronese and Del Pezzo surfaces. The main difficulties and the possible approaches to the case of general surfaces are discussed for and complemented by the example of Bordiga surfaces.

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