A complex surface of general type with , and

Authors:
Heesang Park, Jongil Park and Dongsoo Shin

Journal:
Trans. Amer. Math. Soc. **365** (2013), 5713-5736

MSC (2010):
Primary 14J29; Secondary 14J10, 14J17, 53D05

Published electronically:
January 28, 2013

MathSciNet review:
3091262

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Abstract | References | Similar Articles | Additional Information

Abstract: We construct a new minimal complex surface of general type with , and (in fact, ), which settles the existence question for numerical Campedelli surfaces with all possible algebraic fundamental groups. The main techniques involved in the construction are a rational blow-down surgery and a -Gorenstein smoothing theory.

**1.**I. Bauer, F. Catanese, F. Grunewald, R. Pignatelli,*Quotients of products of curves, new surfaces with and their fundamental groups*. arXiv:0809.3420. To appear in Amer. J. Math.**2.**I. Bauer, F. Catanese, R. Pignatelli,*Surfaces of general type with geometric genus zero: A survey*. arXiv:1004.2583.**3.**I. Bauer, R. Pignatelli,*The classification of minimal product-quotient surfaces with*. arXiv:1006.3209.**4.**F. Catanese,*Babbage’s conjecture, contact of surfaces, symmetric determinantal varieties and applications*, Invent. Math.**63**(1981), no. 3, 433–465. MR**620679**, 10.1007/BF01389064**5.**Hubert Flenner and Mikhail Zaidenberg,*𝐐-acyclic surfaces and their deformations*, Classification of algebraic varieties (L’Aquila, 1992) Contemp. Math., vol. 162, Amer. Math. Soc., Providence, RI, 1994, pp. 143–208. MR**1272698**, 10.1090/conm/162/01532**6.**D. Frapporti,*Mixed surfaces, new surfaces of general type with and their fundamental group*. arXiv:1105.1259.**7.**Masahisa Inoue,*Some new surfaces of general type*, Tokyo J. Math.**17**(1994), no. 2, 295–319. MR**1305801**, 10.3836/tjm/1270127954**8.**J. Keum,*Some new surfaces of general type with*. Unpublished manuscript, 1988.**9.**J. Keum, Y. Lee, H. Park,*Construction of surfaces of general type from elliptic surfaces via -Gorenstein smoothing*. arXiv:1008.1222.**10.**Shigeyuki Kondō,*Enriques surfaces with finite automorphism groups*, Japan. J. Math. (N.S.)**12**(1986), no. 2, 191–282. MR**914299****11.**Yongnam Lee and Jongil Park,*A simply connected surface of general type with 𝑝_{𝑔}=0 and 𝐾²=2*, Invent. Math.**170**(2007), no. 3, 483–505. MR**2357500**, 10.1007/s00222-007-0069-7**12.**Yongnam Lee and Jongil Park,*A complex surface of general type with 𝑝_{𝑔}=0, 𝐾²=2 and 𝐻₁=ℤ/2ℤ*, Math. Res. Lett.**16**(2009), no. 2, 323–330. MR**2496747**, 10.4310/MRL.2009.v16.n2.a9**13.**Margarida Mendes Lopes and Rita Pardini,*Numerical Campedelli surfaces with fundamental group of order 9*, J. Eur. Math. Soc. (JEMS)**10**(2008), no. 2, 457–476. MR**2390332**, 10.4171/JEMS/118**14.**Margarida Mendes Lopes, Rita Pardini, and Miles Reid,*Campedelli surfaces with fundamental group of order 8*, Geom. Dedicata**139**(2009), 49–55. MR**2481836**, 10.1007/s10711-008-9317-2**15.**Daniel Naie,*Numerical Campedelli surfaces cannot have the symmetric group as the algebraic fundamental group*, J. London Math. Soc. (2)**59**(1999), no. 3, 813–827. MR**1709082**, 10.1112/S0024610799007437**16.**Noboru Nakayama,*Zariski-decomposition and abundance*, MSJ Memoirs, vol. 14, Mathematical Society of Japan, Tokyo, 2004. MR**2104208****17.**Jorge Neves and Stavros Argyrios Papadakis,*A construction of numerical Campedelli surfaces with torsion ℤ/6*, Trans. Amer. Math. Soc.**361**(2009), no. 9, 4999–5021. MR**2506434**, 10.1090/S0002-9947-09-04716-3**18.**Heesang Park, Jongil Park, and Dongsoo Shin,*A simply connected surface of general type with 𝑝_{𝑔}=0 and 𝐾²=4*, Geom. Topol.**13**(2009), no. 3, 1483–1494. MR**2496050**, 10.2140/gt.2009.13.1483**19.**R. Pignatelli, A personal communication.**20.**M. Reid,*Surfaces with ,*. Preprint available at`http://www.warwick.ac.uk/~masda/surf/`.**21.**Miles Reid,*Surfaces with 𝑝_{𝑔}=0, 𝐾²=1*, J. Fac. Sci. Univ. Tokyo Sect. IA Math.**25**(1978), no. 1, 75–92. MR**494596****22.**Miles Reid,*Campedelli versus Godeaux*, Problems in the theory of surfaces and their classification (Cortona, 1988), Sympos. Math., XXXII, Academic Press, London, 1991, pp. 309–365. MR**1273384****23.**Gang Xiao,*Surfaces fibrées en courbes de genre deux*, Lecture Notes in Mathematics, vol. 1137, Springer-Verlag, Berlin, 1985 (French). MR**872271**

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Additional Information

**Heesang Park**

Affiliation:
School of Mathematics, Korea Institute for Advanced Study, Seoul 130-722, Korea

Email:
hspark@kias.re.kr

**Jongil Park**

Affiliation:
Department of Mathematical Sciences, Seoul National University, Seoul 151-747, Korea – and – Korea Institute for Advanced Study, Seoul 130-722, Korea

Email:
jipark@snu.ac.kr

**Dongsoo Shin**

Affiliation:
Department of Mathematics, Chungnam National University, Daejeon 305-764, Korea

Email:
dsshin@cnu.ac.kr

DOI:
https://doi.org/10.1090/S0002-9947-2013-05696-6

Keywords:
$\mathbb{Q}$-Gorenstein smoothing,
rational blow-down surgery,
surface of general type

Received by editor(s):
February 21, 2011

Received by editor(s) in revised form:
August 8, 2011, and August 9, 2011

Published electronically:
January 28, 2013

Article copyright:
© Copyright 2013
American Mathematical Society