Surfaces with triple points
Authors:
Stephan Endraß, Ulf Persson and Jan Stevens
Journal:
J. Algebraic Geom. 12 (2003), 367-404
DOI:
https://doi.org/10.1090/S1056-3911-02-00327-2
Published electronically:
November 14, 2002
MathSciNet review:
1949649
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Abstract |
References |
Additional Information
Abstract: In this paper we compute upper bounds for the number of ordinary triple points on a hypersurface in $\mathbb {P}^3$ and give a complete classification for degree six (degree four or less is trivial, and five is elementary). However, the real purpose is to point out the intricate geometry of examples with many triple points, and how it fits with the general classification of surfaces.
[AGV]arnold V. I. Arnol’d, S.M. Gusein-Zade, A.N. Varchenko, Singularities of Differentiable Maps, Volume II, Birkhäuser, Boston (1988)
[BPV]barth W. Barth, C. Peters, A. Van de Ven, Compact Complex Surfaces, Springer, Berlin, 1984.
[B–M]bayer D. Bayer, M. Stillman, Macaulay: A system for computation in algebraic geometry and commutative algebra, Computer software available via anonymous ftp from: ftp://www.math.columbia.edu/pub/bayer/Macaulay/
[C]cayley A. Cayley, A memoir on quartic surfaces, In: Coll. Works VII (1894) 133–181.
[DV]duval P. Du Val, On isolated singularities of surfaces which do not affect the conditions of adjunction I–III, Proc. Camb. Philos. Soc. 30 (1934), 453–491.
[G]gallarati D. Gallarati, Sulle superficie del quinto ordine dotate di punti tripli, Rend. Accad. Naz. Lincei serie VIII, vol. XII (1952) 70–75.
[M]miyaoka Y. Miyaoka, The maximal number of quotient singularities on surfaces with given numerical invariants, Math. Annalen 268 (1984) 159–171.
[R]rohn K. Rohn, Die Flächen vierter Ordnung hinsichtlich ihrer Knotenpunkte und ihrer Gestaltung, S. Hirzel (1886).
[SR]semple J. G. Semple, L. Roth, Introduction to Algebraic Geometry, Oxford University Press (1949).
[Y]yang J.G. Yang, On quintic surfaces of general type, Trans. Amer. Math. Soc. 295 (1986), 431–473.
[W]wahl J. Wahl, Miyaoka-Yau inequality for normal surfaces and local analogues, Contemp. Math. 162 (1994), 381–402.
[AGV]arnold V. I. Arnol’d, S.M. Gusein-Zade, A.N. Varchenko, Singularities of Differentiable Maps, Volume II, Birkhäuser, Boston (1988)
[BPV]barth W. Barth, C. Peters, A. Van de Ven, Compact Complex Surfaces, Springer, Berlin, 1984.
[B–M]bayer D. Bayer, M. Stillman, Macaulay: A system for computation in algebraic geometry and commutative algebra, Computer software available via anonymous ftp from: ftp://www.math.columbia.edu/pub/bayer/Macaulay/
[C]cayley A. Cayley, A memoir on quartic surfaces, In: Coll. Works VII (1894) 133–181.
[DV]duval P. Du Val, On isolated singularities of surfaces which do not affect the conditions of adjunction I–III, Proc. Camb. Philos. Soc. 30 (1934), 453–491.
[G]gallarati D. Gallarati, Sulle superficie del quinto ordine dotate di punti tripli, Rend. Accad. Naz. Lincei serie VIII, vol. XII (1952) 70–75.
[M]miyaoka Y. Miyaoka, The maximal number of quotient singularities on surfaces with given numerical invariants, Math. Annalen 268 (1984) 159–171.
[R]rohn K. Rohn, Die Flächen vierter Ordnung hinsichtlich ihrer Knotenpunkte und ihrer Gestaltung, S. Hirzel (1886).
[SR]semple J. G. Semple, L. Roth, Introduction to Algebraic Geometry, Oxford University Press (1949).
[Y]yang J.G. Yang, On quintic surfaces of general type, Trans. Amer. Math. Soc. 295 (1986), 431–473.
[W]wahl J. Wahl, Miyaoka-Yau inequality for normal surfaces and local analogues, Contemp. Math. 162 (1994), 381–402.
Additional Information
Stephan Endraß
Affiliation:
Micronas GmbH, P. O. Box 840, D 79108 Freiburg, Germany
Email:
endrass@micronas.com
Ulf Persson
Affiliation:
Matematik, Chalmers tekniska högskola, SE 412 96 Göteborg, Sweden
Email:
ulfp@math.chalmers.se
Jan Stevens
Affiliation:
Matematik, Chalmers tekniska högskola, SE 412 96 Göteborg, Sweden
ORCID:
0000-0001-9099-419X
Email:
stevens@math.chalmers.se
Received by editor(s):
November 29, 2000
Published electronically:
November 14, 2002
Additional Notes:
The third author was partially supported by the Swedish Natural Science Research Council (NFR)
