The enumerative geometry of cubic hypersurfaces: point and line conditions

• Belotti, Mara ^[1] ; Danelon, Alessandro ^[2] ; Fevola, Claudia ^[3] ; Kretschmer, Andreas ^[4]
  1. [1] Technical University of Berlin

     Technical University of Berlin

     Berlin, Stadt, Alemania

     
  2. [2] Eindhoven University of Technology

     Eindhoven University of Technology

     Países Bajos

     
  3. [3] Max Planck Institute for Mathematics in the Sciences

     Max Planck Institute for Mathematics in the Sciences

     Kreisfreie Stadt Leipzig, Alemania

     
  4. [4] Fakultät für Mathematik, Institut für Algebra und Geometrie, Otto-von-Guericke-Universität Magdeburg, Universitätsplatz 2, 39106, Magdeburg, Germany

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• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 75, Fasc. 2, 2024, págs. 593-627
• Idioma: inglés
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• Resumen

  • The set of smooth cubic hypersurfaces in Pn is an open subset of a projective space. A compactification of the latter which allows to count the number of smooth cubic hypersurfaces tangent to a prescribed number of lines and passing through a given number of points is termed a 1–complete variety of cubic hypersurfaces, in analogy with the space of complete quadrics. Imitating the work of Aluffi for plane cubic curves, we construct such a space in arbitrary dimensions by a sequence of five blow-ups. The counting problem is then reduced to the computation of five total Chern classes. In the end, we derive the desired numbers in the case of cubic surfaces.