Fano schemes of lines on singular cubic hypersurfaces and their Picard schemes

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Heckel, Tobias: Fano schemes of lines on singular cubic hypersurfaces and their Picard schemes. Hannover : Gottfried Wilhelm Leibniz Universität, Diss., 2020, 98 S. DOI: https://doi.org/10.15488/9921

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Sum total of downloads: 286




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Abstract: 
To every cubic hypersurface $X$ we associate the parameter space of lines contained in $X$; this is called the Fano scheme of lines on $X$ and denoted $F(X)$. If X admits an isolated singular point of ADE-type, we prove that $F(X)$ admits hypersurface singularities of the same type transversally along the regular part of its singular locus. As was shown by H. Clemens and P. Griffiths, the Albanese variety $\operatorname{Alb}(F(X))$ of the Fano scheme of lines $F(X)$ on a smooth cubic threefold $X$ is isomorphic to the intermediate Jacobian $IJ(X)$ of $X$. G. van der Geer and A. Kouvidakis generalised this result to nodal cubic threefolds, replacing the Albanese variety $\operatorname{Alb}(F(X))$ by the Picard scheme $\operatorname{Pic}^0(F(X))$.We study more generally degenerations of the Picard scheme $\operatorname{Pic}^0(F(X))$ when the smooth cubic threefold $X$ degenerates to a cubic threefold with unique singular point of type $A_k$ and prove that these degenerations define points in Mumford’s partial compactification $\mathcal{A}_5'$ of the moduli space $\mathcal{A}_5$ of principally polarised Abelian varieties of genus five.
License of this version: CC BY 3.0 DE
Document Type: doctoralThesis
Publishing status: publishedVersion
Issue Date: 2020
Appears in Collections:Fakultät für Mathematik und Physik
Dissertationen

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pos. country downloads
total perc.
1 image of flag of Germany Germany 157 54.90%
2 image of flag of Italy Italy 35 12.24%
3 image of flag of United States United States 22 7.69%
4 image of flag of United Kingdom United Kingdom 10 3.50%
5 image of flag of India India 9 3.15%
6 image of flag of No geo information available No geo information available 7 2.45%
7 image of flag of Greece Greece 6 2.10%
8 image of flag of China China 5 1.75%
9 image of flag of Switzerland Switzerland 5 1.75%
10 image of flag of Singapore Singapore 4 1.40%
    other countries 26 9.09%

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