2023/05/03 by Mara Belotti, Alessandro Danelon, Claudia Fevola +1
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Algorithm #Combinatorics #Complex projective space #Computation #Cubic function #Cubic surface #Geometric and Algebraic Topology #Geometry #Mathematical analysis #Mathematics #Polynomial and algebraic computation #Projective space #Pure mathematics #Real projective plane #Tangent #Tangent space
paper · pdf · doi:10.1007/s13348-023-00401-z
crossref issued 2023/05/03 · crossref published 2023/05/03 · crossref published-online 2023/05/03 · openalex publication_date 2023/05/03 · crossref created 2023/05/03 · crossref deposited 2024/03/27 · crossref published-print 2024/05/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06 · crossref indexed 2026/08/06
Abstract The set of smooth cubic hypersurfaces in \mathbb Pn <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math> 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.