Moduli of algebraic hypersurfaces via homotopy principles
Research output: Book/Report › Ph.D. thesis › Research
Standard
Moduli of algebraic hypersurfaces via homotopy principles. / Aumonier, Alexis.
Department of Mathematical Sciences, Faculty of Science, University of Copenhagen, 2023. 137 p.Research output: Book/Report › Ph.D. thesis › Research
Harvard
APA
Vancouver
Author
Bibtex
}
RIS
TY - BOOK
T1 - Moduli of algebraic hypersurfaces via homotopy principles
AU - Aumonier, Alexis
PY - 2023
Y1 - 2023
N2 - In this thesis, I prove a general h-principle for algebraic sections of vector bundles, and use it to investigate the homology of moduli spaces of smooth algebraic hypersurfaces. The thesis consists of an introduction followed by three papers, the last of which is joint with Ronno Das.In the first paper, I consider spaces of algebraic sections of vector bundles subject to differential relations. On smooth projective complex varieties, I prove that the homology of such a space coincides in a range with that of a space of continuous sections of an associated bundle. As an immediate consequence, I show stability of the rational cohomology for complement of discriminants in linear systems of hypersurfaces of increasing degree. This paper is the most technical and its results are used repeatedly throughout the thesis.In the second paper, I study the locus of smooth hypersurfaces inside the Hilbert scheme of a smooth complex projective variety. Using the results of the first paper, I show how part of its cohomology can be computed via an h-principle akin to a scanning map. I also explain how to compare the rational cohomology to that of classifying spaces of diffeomorphisms groups of hypersurfaces.In the third paper, Ronno Das and I study the cohomology of the universal smooth hypersurface bundle with marked points. We adapt the arguments of the first paper to show another h-principle. Using rational models, we deduce rational homological stability for this space.
AB - In this thesis, I prove a general h-principle for algebraic sections of vector bundles, and use it to investigate the homology of moduli spaces of smooth algebraic hypersurfaces. The thesis consists of an introduction followed by three papers, the last of which is joint with Ronno Das.In the first paper, I consider spaces of algebraic sections of vector bundles subject to differential relations. On smooth projective complex varieties, I prove that the homology of such a space coincides in a range with that of a space of continuous sections of an associated bundle. As an immediate consequence, I show stability of the rational cohomology for complement of discriminants in linear systems of hypersurfaces of increasing degree. This paper is the most technical and its results are used repeatedly throughout the thesis.In the second paper, I study the locus of smooth hypersurfaces inside the Hilbert scheme of a smooth complex projective variety. Using the results of the first paper, I show how part of its cohomology can be computed via an h-principle akin to a scanning map. I also explain how to compare the rational cohomology to that of classifying spaces of diffeomorphisms groups of hypersurfaces.In the third paper, Ronno Das and I study the cohomology of the universal smooth hypersurface bundle with marked points. We adapt the arguments of the first paper to show another h-principle. Using rational models, we deduce rational homological stability for this space.
M3 - Ph.D. thesis
BT - Moduli of algebraic hypersurfaces via homotopy principles
PB - Department of Mathematical Sciences, Faculty of Science, University of Copenhagen
ER -
ID: 376982674