A construction of numerical Campedelli surfaces with torsion $\mathbb {Z}/6$
HTML articles powered by AMS MathViewer
- by Jorge Neves and Stavros Argyrios Papadakis PDF
- Trans. Amer. Math. Soc. 361 (2009), 4999-5021 Request permission
Abstract:
We produce a family of numerical Campedelli surfaces with $\mathbb {Z}/6$ torsion by constructing the canonical ring of the étale 6 to 1 cover using serial unprojection. In Section 2 we develop the necessary algebraic machinery. Section 3 contains the numerical Campedelli surface construction, while Section 4 contains remarks and open questions.References
- Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR 1251956
- Wolf P. Barth, Klaus Hulek, Chris A. M. Peters, and Antonius Van de Ven, Compact complex surfaces, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 4, Springer-Verlag, Berlin, 2004. MR 2030225, DOI 10.1007/978-3-642-57739-0
- Brown, G., Graded ring database homepage, online searchable database, available from http://pcmat12.kent.ac.uk/grdb/index.php
- Winfried Bruns and Udo Vetter, Determinantal rings, Lecture Notes in Mathematics, vol. 1327, Springer-Verlag, Berlin, 1988. MR 953963, DOI 10.1007/BFb0080378
- Alessio Corti and Miles Reid, Weighted Grassmannians, Algebraic geometry, de Gruyter, Berlin, 2002, pp. 141–163. MR 1954062
- Igor Dolgachev, Weighted projective varieties, Group actions and vector fields (Vancouver, B.C., 1981) Lecture Notes in Math., vol. 956, Springer, Berlin, 1982, pp. 34–71. MR 704986, DOI 10.1007/BFb0101508
- David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960, DOI 10.1007/978-1-4612-5350-1
- H. Flenner, L. O’Carroll, and W. Vogel, Joins and intersections, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1999. MR 1724388, DOI 10.1007/978-3-662-03817-8
- Frantzen, Kr., On K$3$-surfaces in weighted projective space. Univ. of Warwick M.Sc. thesis, Sep 2004 v+55 pp., available from http://pcmat12.kent.ac.uk/grdb/Doc/papers.php
- Greuel, G.-M, Pfister, G., and Schönemann, H., Singular 2.0. A Computer Algebra System for Polynomial Computations. Centre for Computer Algebra, University of Kaiserslautern (2001), available from http://www.singular.uni-kl.de
- Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157, DOI 10.1007/978-1-4757-3849-0
- A. R. Iano-Fletcher, Working with weighted complete intersections, Explicit birational geometry of 3-folds, London Math. Soc. Lecture Note Ser., vol. 281, Cambridge Univ. Press, Cambridge, 2000, pp. 101–173. MR 1798982
- Andrew R. Kustin and Matthew Miller, Constructing big Gorenstein ideals from small ones, J. Algebra 85 (1983), no. 2, 303–322. MR 725084, DOI 10.1016/0021-8693(83)90096-0
- Yongnam Lee and Jongil Park, A simply connected surface of general type with $p_g=0$ and $K^2=2$, Invent. Math. 170 (2007), no. 3, 483–505. MR 2357500, DOI 10.1007/s00222-007-0069-7
- Qing Liu, Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Mathematics, vol. 6, Oxford University Press, Oxford, 2002. Translated from the French by Reinie Erné; Oxford Science Publications. MR 1917232
- 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, DOI 10.4171/JEMS/118
- 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, DOI 10.1112/S0024610799007437
- Papadakis, S., Gorenstein rings and Kustin–Miller unprojection, Univ. of Warwick Ph.D. thesis, Aug 2001, vi + 72 pp., available from http:// www. math. ist. utl. pt/ $\sim$papadak/
- Stavros Argyrios Papadakis, Kustin-Miller unprojection with complexes, J. Algebraic Geom. 13 (2004), no. 2, 249–268. MR 2047698, DOI 10.1090/S1056-3911-03-00350-3
- Stavros Argyrios Papadakis, Type II unprojection, J. Algebraic Geom. 15 (2006), no. 3, 399–414. MR 2219843, DOI 10.1090/S1056-3911-06-00433-4
- Stavros Argyrios Papadakis and Miles Reid, Kustin-Miller unprojection without complexes, J. Algebraic Geom. 13 (2004), no. 3, 563–577. MR 2047681, DOI 10.1090/S1056-3911-04-00343-1
- Reid, M., Graded Rings and Birational Geometry, in Proc. of algebraic symposium (Kinosaki, Oct 2000), K. Ohno (Ed.) 1–72, available from www. maths. warwick. ac. uk/ $\sim$miles/3folds
- 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
- Reid, M., Examples of type IV unprojection, preprint, math.AG/0108037, 16 pp.
Additional Information
- Jorge Neves
- Affiliation: Centre for Mathematics, University of Coimbra, 3001-454 Coimbra, Portugal
- Email: neves@mat.uc.pt
- Stavros Argyrios Papadakis
- Affiliation: Center for Mathematical Analysis, Geometry, and Dynamical Systems, Departamento de Matemática, Instituto Superior Técnico, Universidade Técnica de Lisboa, 1049-001 Lisboa, Portugal
- Email: papadak@math.ist.utl.pt
- Received by editor(s): April 13, 2007
- Received by editor(s) in revised form: December 3, 2007
- Published electronically: April 15, 2009
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 361 (2009), 4999-5021
- MSC (2000): Primary 14J29; Secondary 13H10, 14M05
- DOI: https://doi.org/10.1090/S0002-9947-09-04716-3
- MathSciNet review: 2506434