Abstract
We give a formula for the Hodge numbers of a three-dimensional hypersurface in a weighted projective space with only ordinary triple points as singularities.
Funding statement: This research was partially supported by the National Science Center grant no. 2014/13/B/ST1/00133.
Acknowledgements
I would like to thank Remke Kloosterman for helpful discussions on this topic.
Communicated by: I. Coskun
References
[1] C. H. Clemens, Double solids. Adv. in Math. 47 (1983), 107–230. MR690465 Zbl 0509.1404510.1016/0001-8708(83)90025-7Search in Google Scholar
[2] S. Cynk, S. Rams, Defect via differential forms with logarithmic poles. Math. Nachr. 284 (2011), 2148–2158. MR2859755 Zbl 1239.1403110.1002/mana.200910220Search in Google Scholar
[3] I. Dolgachev, Weighted projective varieties. In: Group actions and vector fields (Vancouver, B. C., 1981), volume 956 of Lecture Notes in Math., 34–71, Springer 1982. MR704986 Zbl 0516.1401410.1007/BFb0101508Search in Google Scholar
[4] S. Endrass, U. Persson, J. Stevens, Surfaces with triple points. J. Algebraic Geom. 12 (2003), 367–404. MR1949649 Zbl 1097.1403010.1090/S1056-3911-02-00327-2Search in Google Scholar
[5] H. Esnault, E. Viehweg, Lectures on vanishing theorems, volume 20 of DMV Seminar. Birkhäuser Verlag, Basel 1992. MR1193913 Zbl 0779.1400310.1007/978-3-0348-8600-0Search in Google Scholar
[6] R. Kloosterman, S. Rams, Quintic threefolds with triple points. Preprint 2017, arXiv:1712.05710 [math.AG]Search in Google Scholar
[7] C. Peters, J. Steenbrink, Infinitesimal variations of Hodge structure and the generic Torelli problem for projective hypersurfaces (after Carlson, Donagi, Green, Griffiths, Harris). In: Classification of algebraic and analytic manifolds (Katata, 1982), 399–463, Birkhäuser Boston, Boston, MA 1983. MR728615 Zbl 0523.14009Search in Google Scholar
[8] J. Stevens, Sextic surfaces with 10 triple points. In: Singularities and computer algebra, volume 324 of London Math. Soc. Lecture Note Ser., 315–331, Cambridge Univ. Press 2006. MR2228237 Zbl 1105.1405210.1017/CBO9780511526374.016Search in Google Scholar
[9] D. van Straten, A quintic hypersurface in ℙ4 with 130 nodes. Topology 32 (1993), 857–864. MR1241876 Zbl 0801.1401510.1016/0040-9383(93)90054-YSearch in Google Scholar
[10] A. N. Varchenko, Semicontinuity of the spectrum and an upper bound for the number of singular points of the projective hypersurface. Dokl. Akad. Nauk SSSR 270 (1983), 1294–1297. English translation: Soviet Math. Dokl. 27 (1983), no. 3, 735–739. MR712934 Zbl 0537.14003Search in Google Scholar
[11] J. Werner, Kleine Auflösungen spezieller dreidimensionaler Varietäten, volume 186 of Bonner Mathematische Schriften. Universität Bonn, Mathematisches Institut, 1987. MR930270 Zbl 0657.14021Search in Google Scholar
© 2021 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Complex geodesics in convex domains and ℂ-convexity of semitube domains
- Triharmonic Riemannian submersions from 3-dimensional space forms
- Ricci almost solitons and contact geometry
- Shapes of centrally symmetric octahedra with prescribed cone-deficits
- Equivariant Ulrich bundles on exceptional homogeneous varieties
- Exotic Steiner chains in Miquelian Möbius planes of odd order
- Double cover K3 surfaces of Hirzebruch surfaces
- The dual cone of sums of non-negative circuit polynomials
- Einstein tori and crooked surfaces
- Compact null hypersurfaces in Lorentzian manifolds
- Hyperbolic torsion polynomials of pretzel knots
- Geodesic orbit Randers metrics on spheres
- Rational conic fibrations of sectional genus two
- Hodge numbers of hypersurfaces in ℙ4 with ordinary triple points
Articles in the same Issue
- Frontmatter
- Complex geodesics in convex domains and ℂ-convexity of semitube domains
- Triharmonic Riemannian submersions from 3-dimensional space forms
- Ricci almost solitons and contact geometry
- Shapes of centrally symmetric octahedra with prescribed cone-deficits
- Equivariant Ulrich bundles on exceptional homogeneous varieties
- Exotic Steiner chains in Miquelian Möbius planes of odd order
- Double cover K3 surfaces of Hirzebruch surfaces
- The dual cone of sums of non-negative circuit polynomials
- Einstein tori and crooked surfaces
- Compact null hypersurfaces in Lorentzian manifolds
- Hyperbolic torsion polynomials of pretzel knots
- Geodesic orbit Randers metrics on spheres
- Rational conic fibrations of sectional genus two
- Hodge numbers of hypersurfaces in ℙ4 with ordinary triple points