A note on degenerations of del Pezzo surfaces

Prokhorov, Yuri

doi:10.5802/aif.2934

A note on degenerations of del Pezzo surfaces
[Une remarque sur dégénérescences de surfaces de del Pezzo]

Prokhorov, Yuri ^1,^2,³

¹ Steklov Institute of Mathematics 8 Gubkina street Moscow 119991 (Russia)
² Department of Algebra, Faculty of Mathematics Moscow State University Moscow 117234 (Russia)
³ Laboratory of Algebraic Geometry, National Research University Higher School of Economics, 7 Vavilova Str. Moscow 117312 (Russia)

Annales de l'Institut Fourier, Tome 65 (2015) no. 1, pp. 369-388.

Résumé
Abstract

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.

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

Zbl

DOI : 10.5802/aif.2934

Classification : 14J10, 14E30
Keywords: del Pezzo surface, T-singularity, deformation
Mot clés : surface de Del Pezzo, T-singularité, déformation

Affiliations des auteurs :

Prokhorov, Yuri ^{1, 2, 3}

@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 = {http://www.numdam.org/articles/10.5802/aif.2934/}
}

TY  - JOUR
AU  - Prokhorov, Yuri
TI  - A note on degenerations of del Pezzo surfaces
JO  - Annales de l'Institut Fourier
PY  - 2015
SP  - 369
EP  - 388
VL  - 65
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2934/
DO  - 10.5802/aif.2934
LA  - en
ID  - AIF_2015__65_1_369_0
ER  -

%0 Journal Article
%A Prokhorov, Yuri
%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 http://www.numdam.org/articles/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, Tome 65 (2015) no. 1, pp. 369-388. doi : 10.5802/aif.2934. http://www.numdam.org/articles/10.5802/aif.2934/

Bibliographie
Cité par

[1] Brieskorn, Egbert Rationale Singularitäten komplexer Flächen, Invent. Math., Volume 4 (1967/1968), pp. 336-358 | DOI | MR | Zbl

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

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

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

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

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

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

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

[9] Kollár, János Flips and abundance for algebraic threefolds, 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

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

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

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

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

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

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

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

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

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

[19] Wahl, Jonathan Smoothings of normal surface singularities, Topology, Volume 20 (1981) no. 3, pp. 219-246 | DOI | MR | Zbl

Cité par Sources :