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arXiv · 2201.05424

The enumerative geometry of cubic hypersurfaces: point and line conditions

Mara Belotti·Alessandro Danelon·Claudia Fevola·Andreas Kretschmer
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Abstract

In order to count the number of smooth cubic hypersurfaces tangent to a prescribed number of lines and passing through a given number of points, we construct a compactification of their moduli space. We term the latter a $1$--\textit{complete variety of cubic hypersurfaces} in analogy to the space of complete quadrics. Paolo Aluffi explored the case of plane cubic curves. Starting from his work, we construct such a space in arbitrary dimension by a sequence of five blow-ups. The counting problem is then reduced to the computation of five Chern classes, climbing the sequence of blow-ups. Computing the last of these is difficult due to the fact that the vector bundle is not given explicitly. Identifying a restriction of this vector bundle, we arrive at the desired numbers in the case of cubic surfaces.

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Mara Belotti, Alessandro Danelon, Claudia Fevola, Andreas Kretschmer. 2022-01-14. The enumerative geometry of cubic hypersurfaces: point and line conditions. https://arxiv.org/abs/2201.05424

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