Tohoku Mathematical Journal

All toric local complete intersection singularities admit projective crepant resolutions

Dimitrios I. Dais, Christian Haase, and Günter M. Ziegler

Full-text: Open access


It is known that the underlying spaces of all abelian quotient singularities which are embeddable as complete intersections of hypersurfaces in an affine space can be overall resolved by means of projective torus-equivariant crepant birational morphisms in all dimensions. In the present paper we extend this result to the entire class of toric local complete intersection singularities. Our strikingly simple proof makes use of Nakajima's classification theorem and of some techniques from toric and discrete geometry.

Article information

Tohoku Math. J. (2), Volume 53, Number 1 (2001), 95-107.

First available in Project Euclid: 3 May 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14M25: Toric varieties, Newton polyhedra [See also 52B20]
Secondary: 14E15: Global theory and resolution of singularities [See also 14B05, 32S20, 32S45]


Dais, Dimitrios I.; Haase, Christian; Ziegler, Günter M. All toric local complete intersection singularities admit projective crepant resolutions. Tohoku Math. J. (2) 53 (2001), no. 1, 95--107. doi:10.2748/tmj/1178207533.

Export citation


  • [B] V. V. BATYREv, Non-Archimedian integrals and stringy Euler numbers of log-terminal pairs, J. European Math. Soc.1 (1999), 5-33.
  • [BD] V. V. BATYREV AND D. I. DAIS, Strong McKay correspondence, string-theoretic Hodge numbers an mirror symmetry, Topology 35 (1996), 901-929.
  • [BGT] W. BRUNS, J. GUBELADZE AND N. V. TRUNG, Normal polytopes, triangulations and Koszul algebras, J. Reine Angew. Math. 485 (1997), 123-160.
  • [DHaZ] D. I. DAIS, C. HAASE AND G. M. ZIEGLER, All toric l.c.i.-singularities admit projective crepant reso lutions, TU-Berlin Preprint 614/1998 and LANL-Preprint alg-geom/9812025, 33 pages.
  • [DHH] D. I. DAIS, U. -U. HAUS AND M. HENK, On crepant resolutions of 2-arameter series of Gorenstei cyclic quotient singularities, Results Math. 33 (1998), 208-265.
  • [DH] D. I. DAIS AND M. HENK, On a series of Gorenstein cyclic quotient singularities admitting a uniqu projective crepant resolution, alg-geom/9803094; to appear in: Combinatorial Convex Geometry and Toric Varieties (ed. by G. Ewald & B. Teissier), Birkhauser, Basel.
  • [DHZ98] D. I. DAIS, M. HENK AND G. M. ZIEGLER, All abelian quotient c.i.-singularities admit projectiv crepant resolutions in all dimensions, Adv. in Math. 139 (1998), 192-239.
  • [DHZ99] D. I. DAIS, M. HENK AND G. M. ZIEGLER, On the existence of crepant resolutions of Gorenstei abelian quotient singularities in dimensions 4, in preparation.
  • [E] G. EWALD, Combinatorial Convexity and Algebraic Geometry, Grad. Texts in Math. 168, Springer Verlag, New York, 1996.
  • [F] W. FULTON, Introduction to Toric Varieties, Ann. of Math. Stud. 131, Princeton Univ. Press, Princeton, 1993.
  • [IR] Y. ITO AND M. REID, The McKay correspondence for finite subgroups of SL(3, C), Higher Dimensiona Complex Varieties (Trento, 1994), 221-240, de Gruyter, Berlin 1996.
  • [KKMS] G. KEMPF, F. KNUDSEN, D. MUMFORD AND D. SAINT-DONAT, Toroidal Embeddings I, Lecture Note in Math. 339, Springer-Verlag, Berlin-New York, 1973.
  • [L] C. W. LEE, Regular triangulations of convex polytopes, Applied Geometry and Discrete Mathematics-The Victor Klee Festschrift, 443-456, DIMACS Ser.Discrete Math. Theoret. Comput. Sci. 4, Amer. Math. Soc., Providence, RI, 1991.
  • [N] H. NAKAJIMA, Affine torus embeddings which are complete intersections, Thoku Math. J. 38 (1986), 85-98.
  • [O] T. ODA, Convex Bodies and Algebraic Geometry, An Introduction to the Theory of Toric Varieties, Ergeb. Math. Grenzgeb. (3) 15, Springer-Verlag, Berlin-New York, 1988.
  • [R] M. REID, McKay correspondence, Algebraic Geometry (Kinosaki, 1996), 14-41
  • [W] K. WATANABE, Invariant subrings which are complete intersections I, (Invariantsubrings of finite Abelia groups), Nagoya Math. J. 77 (1980), 89-98.