A note on degenerations of del Pezzo surfaces
Annales de l'Institut Fourier, Volume 65 (2015) no. 1, pp. 369-388.

We prove that for a Q-Gorenstein degeneration X of del Pezzo surfaces, the number of non-Du Val singularities is at most ρ(X)+2. Degenerations with ρ(X)+2 and ρ(X)+1 non-Du Val points are investigated

Nous montrons que pour une dégénérescence Q-Gorenstein X de surfaces de del Pezzo, le nombre de singularités non-Du Val est au plus ρ(X)+2. Les dégénérescences avec ρ(X)+2 et ρ(X)+1 points non-Du Val sont étudiées.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.2934
Classification: 14J10,  14E30
Keywords: del Pezzo surface, T-singularity, deformation
@article{AIF_2015__65_1_369_0,
     author = {Prokhorov, Yuri},
     title = {A note on degenerations of del {Pezzo} surfaces},
     journal = {Annales de l'Institut Fourier},
     pages = {369--388},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {65},
     number = {1},
     year = {2015},
     doi = {10.5802/aif.2934},
     zbl = {06496543},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2934/}
}
TY  - JOUR
TI  - A note on degenerations of del Pezzo surfaces
JO  - Annales de l'Institut Fourier
PY  - 2015
DA  - 2015///
SP  - 369
EP  - 388
VL  - 65
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2934/
UR  - https://zbmath.org/?q=an%3A06496543
UR  - https://doi.org/10.5802/aif.2934
DO  - 10.5802/aif.2934
LA  - en
ID  - AIF_2015__65_1_369_0
ER  - 
%0 Journal Article
%T A note on degenerations of del Pezzo surfaces
%J Annales de l'Institut Fourier
%D 2015
%P 369-388
%V 65
%N 1
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.2934
%R 10.5802/aif.2934
%G en
%F AIF_2015__65_1_369_0
Prokhorov, Yuri. A note on degenerations of del Pezzo surfaces. Annales de l'Institut Fourier, Volume 65 (2015) no. 1, pp. 369-388. doi : 10.5802/aif.2934. https://aif.centre-mersenne.org/articles/10.5802/aif.2934/

[1] Brieskorn, Egbert Rationale Singularitäten komplexer Flächen, Invent. Math., Tome 4 (1967/1968), pp. 336-358 | Article | MR: 222084 | Zbl: 0219.14003

[2] Durfee, Alan H. Fifteen characterizations of rational double points and simple critical points, Enseign. Math. (2), Tome 25 (1979) no. 1-2, pp. 131-163 | MR: 543555 | Zbl: 0418.14020

[3] Hacking, Paul Compact moduli of plane curves, Duke Math. J., Tome 124 (2004) no. 2, pp. 213-257 | Article | MR: 2078368 | Zbl: 1060.14034

[4] Hacking, Paul Compact moduli spaces of surfaces of general type, Compact moduli spaces and vector bundles (Contemp. Math.) Tome 564, Amer. Math. Soc., Providence, RI, 2012, pp. 1-18 | Article | MR: 2895182 | Zbl: 1254.14043

[5] Hacking, Paul Exceptional bundles associated to degenerations of surfaces, Duke Math. J., Tome 162 (2013) no. 6, pp. 1171-1202 | Article | MR: 3053568 | Zbl: 1282.14074

[6] Hacking, Paul; Prokhorov, Yuri Smoothable del Pezzo surfaces with quotient singularities, Compos. Math., Tome 146 (2010) no. 1, pp. 169-192 | Article | MR: 2581246 | Zbl: 1194.14054

[7] Karpov, B. V.; Nogin, D. Yu. Three-block exceptional sets on del Pezzo surfaces, Izv. Ross. Akad. Nauk Ser. Mat., Tome 62 (1998) no. 3, pp. 3-38 | Article | MR: 1642152 | Zbl: 0949.14026

[8] Kollár, J.; Shepherd-Barron, N. I. Threefolds and deformations of surface singularities, Invent. Math., Tome 91 (1988) no. 2, pp. 299-338 | Article | MR: 922803 | Zbl: 0642.14008

[9] Flips and abundance for algebraic threefolds (Kollár, János, ed.), Société Mathématique de France, Paris, 1992, pp. 1-258 (Papers from the Second Summer Seminar on Algebraic Geometry held at the University of Utah, Salt Lake City, Utah, August 1991, Astérisque No. 211 (1992)) | MR: 1225842

[10] Kollár, János; Mori, Shigefumi Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, Tome 134, Cambridge University Press, Cambridge, 1998, viii+254 pages (With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original) | Article | MR: 1658959 | Zbl: 0926.14003

[11] Manetti, Marco Normal degenerations of the complex projective plane, J. Reine Angew. Math., Tome 419 (1991), pp. 89-118 | Article | MR: 1116920 | Zbl: 0719.14023

[12] Miyanishi, M.; Zhang, D.-Q. Gorenstein log del Pezzo surfaces of rank one, J. Algebra, Tome 118 (1988) no. 1, pp. 63-84 | Article | MR: 961326 | Zbl: 0664.14019

[13] Morrison, David R. The birational geometry of surfaces with rational double points, Math. Ann., Tome 271 (1985) no. 3, pp. 415-438 | Article | MR: 787190 | Zbl: 0539.14008

[14] Prokhorov, Yu. G. On semistable Mori contractions, Izv. Ross. Akad. Nauk Ser. Mat., Tome 68 (2004) no. 2, pp. 147-158 | Article | MR: 2058003 | Zbl: 1075.14014

[15] Prokhorov, Yu. G.; Shokurov, V. V. Towards the second main theorem on complements, J. Algebraic Geom., Tome 18 (2009) no. 1, pp. 151-199 | Article | MR: 2448282 | Zbl: 1159.14020

[16] Prokhorov, Yuri G. Lectures on complements on log surfaces, MSJ Memoirs, Tome 10, Mathematical Society of Japan, Tokyo, 2001, viii+130 pages | MR: 1830440 | Zbl: 1037.14003

[17] Reid, Miles Young person’s guide to canonical singularities, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) (Proc. Sympos. Pure Math.) Tome 46, Amer. Math. Soc., Providence, RI, 1987, pp. 345-414 | MR: 927963 | Zbl: 0634.14003

[18] Shokurov, V. V. Complements on surfaces, J. Math. Sci. (New York), Tome 102 (2000) no. 2, pp. 3876-3932 (Algebraic geometry, 10) | Article | MR: 1794169 | Zbl: 1177.14078

[19] Wahl, Jonathan Smoothings of normal surface singularities, Topology, Tome 20 (1981) no. 3, pp. 219-246 | Article | MR: 608599 | Zbl: 0484.14012

Cited by Sources: