Hostname: page-component-745bb68f8f-5r2nc Total loading time: 0 Render date: 2025-02-11T06:41:09.711Z Has data issue: false hasContentIssue false
  • English
  • Français

Smoothings of Fano Varieties With Normal Crossing Singularities

Published online by Cambridge University Press:  10 August 2015

Nikolaos Tziolas
Nikolaos Tziolas*
Affiliation:
Department of Mathematics, University of Cyprus, PO Box 20537 Nicosia 1678, Cyprus (tziolas@ucy.ac.cy)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper obtains criteria for a Fano variety X defined over an algebraically closed field of characteristic zero with normal crossing singularities to be smoothable. In particular, we show that X is smoothable by a flat deformation X → Δ with smooth total space X if and only if where D is the singular locus of X.

Keywords

algebraic geometryFano varietiessmoothingsnormal crossingPrimary 14D1514D0614J45
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 58 , Issue 3 , October 2015 , pp. 787 - 806
Copyright
Copyright © Edinburgh Mathematical Society 2015 

References

1.Akizuki, Y. and Nakano, S., Note on Kodaira-Spencer’s proof of Lefschetz’s theorem, Proc. Jpn Acad. A 30 (1954), 266–272.Google Scholar
2.Artin, M., Algebraic approximation of structures over complete local rings, Publ. Math. IHES 36 (1969), 23–58.Google Scholar
3.Esnault, E. and Viehweg, E., Lectures on vanishing theorems (Birkhäuser, 1992).CrossRefGoogle Scholar
4.Friedman, R., Global smoothings of varieties with normal crossings, Annals Math. 118 (1983), 75–114.Google Scholar
5.Fujita, T., On Del Pezzo fibrations over curves, Osaka J. Math. 27 (1990), 229–245.Google Scholar
6.Kempf, G., Knudsen, F., Mumford, D. and Saint-Donat, B., Toroidal embeddings I, Lecture Notes in Mathematics, Volume 339 (Springer, 1973).Google Scholar
7.Kachi, Y., Global smoothings of degenerate del Pezzo surfaces with normal crossings, J. Alg. 307 (2007), 249–253.Google Scholar
8.Kato, K., Logarithmic structures of Fontaine-Illusie, in Algebraic analysis geometry and number theory, pp. 191–224 (Johns Hopkins University Press, Baltimore, MD, 1988).Google Scholar
9.Kato, F., Log smooth deformation theory, Tohoku Math. J. 48 (1996), 317–354.Google Scholar
10.Kawamata, Y. and Namikawa, Y., Logarithmic deformations of normal crossing varieties and smoothing of degenerate Calabi-Yau varieties, Invent. Math. 118 (1994), 395–409.CrossRefGoogle Scholar
11.Pinkham, H. and Persson, U., Some examples of nonsmoothable varieties with normal crossings, Duke Math. J. 50 (1983), 477–486.Google Scholar
12.Schlessinger, M., Functors of Artin rings, Trans. Am. Math. Soc. 130 (1968), 208–222.Google Scholar
13.Sernesi, E., Deformations of algebraic schemes (Springer, 2006).Google Scholar
14.Tziolas, N., Q-Gorenstein smoothings of nonnormal surfaces, Am. J. Math. 131 (2009)(1), 171–193.CrossRefGoogle Scholar
15.Tziolas, N., Smoothings of schemes with nonisolated singularities, Michigan Math. J. 59(1 (2010), 25–84.Google Scholar