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 Trudy Mat. Inst. Steklova, 1999, Volume 225, Pages 132–152 (Mi tm716)

How to Calculate Homology Groups of Spaces of Nonsingular Algebraic Projective Hypersurfaces

V. A. Vassiliev

Abstract: A general method of computing cohomology groups of the space of nonsingular algebraic hypersurfaces of degree $d$ in $\mathbf{CP}^n$ is described. Using this method, rational cohomology groups of such spaces with $n=2$, $d\le 4$ and $n=3=d$ and also of the space of nondegenerate quadratic vector fields in $\mathbf C^3$ are calculated.

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English version:
Proceedings of the Steklov Institute of Mathematics, 1999, 225, 121–140

Bibliographic databases:
UDC: 515.14+512.7

Citation: V. A. Vassiliev, “How to Calculate Homology Groups of Spaces of Nonsingular Algebraic Projective Hypersurfaces”, Solitons, geometry, and topology: on the crossroads, Collection of papers dedicated to the 60th anniversary of academician Sergei Petrovich Novikov, Trudy Mat. Inst. Steklova, 225, Nauka, MAIK «Nauka/Inteperiodika», M., 1999, 132–152; Proc. Steklov Inst. Math., 225 (1999), 121–140

Citation in format AMSBIB
\Bibitem{Vas99}
\by V.~A.~Vassiliev
\paper How to Calculate Homology Groups of Spaces of Nonsingular Algebraic Projective Hypersurfaces
\inbook Solitons, geometry, and topology: on the crossroads
\bookinfo Collection of papers dedicated to the 60th anniversary of academician Sergei Petrovich Novikov
\serial Trudy Mat. Inst. Steklova
\yr 1999
\vol 225
\pages 132--152
\publ Nauka, MAIK «Nauka/Inteperiodika»
\mathnet{http://mi.mathnet.ru/tm716}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1725936}
\zmath{https://zbmath.org/?q=an:0981.55008}
\transl
\jour Proc. Steklov Inst. Math.
\yr 1999
\vol 225
\pages 121--140

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. Das R., “The Space of Cubic Surfaces Equipped With a Line”, Math. Z.
2. Vassiliev V.A., “Homology of spaces of knots in any dimensions”, Philosophical Transactions of the Royal Society of London Series A–Mathematical Physical and Engineering Sciences, 359:1784 (2001), 1343–1364
3. Vassiliev V., “Resolutions of discriminants and topology of their complements”, New Developments in Singularity Theory, NATO Science Series, Series II: Mathematics, Physics and Chemistry, 21, 2001, 87–115
4. C. A. M. Peters, J. H. M. Steenbrink, “Degeneration of the Leray spectral sequence for certain geometric quotients”, Mosc. Math. J., 3:3 (2003), 1085–1095
5. V. A. Vassiliev, “Spaces of Hermitian operators with simple spectra and their finite-order cohomology”, Mosc. Math. J., 3:3 (2003), 1145–1165
6. Gorinov A.G., “Conical resolutions of discriminant varieties and real cohomology of the space of nonsingular complex plane projective quintics”, Doklady Mathematics, 67:2 (2003), 259–262
7. Tommasi O., “Rational cohomology of the moduli space of genus 4 curves”, Compositio Mathematica, 141:2 (2005), 359–384
8. de Bobadilla J.F., “Moduli spaces of polynomials in two variables”, Memoirs of the American Mathematical Society, 173:817 (2005), VII
9. Tommasi O., “Rational cohomology of M–3,M–2”, Compositio Mathematica, 143:4 (2007), 986–1002
10. Bergstrom J., Tommasi O., “The rational cohomology of (M)over–bar(4)”, Mathematische Annalen, 338:1 (2007), 207–239
11. V. A. Vassiliev, “Rational homology of the order complex of zero sets of homogeneous quadratic polynomial systems in $\mathbb R^3$”, Proc. Steklov Inst. Math., 290:1 (2015), 197–209
12. V. A. Vassiliev, “Homology groups of spaces of non-resultant quadratic polynomial systems in ${\mathbb R}^3$”, Izv. Math., 80:4 (2016), 791–810
13. Gomez-Gonzales C., “Spaces of Non-Degenerate Maps Between Complex Projective Spaces”, Res. Math. Sci., 7:3 (2020), 26
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