On the equations defining toric l.c.i.-singularities

Authors:
Dimitrios I. Dais and Martin Henk

Journal:
Trans. Amer. Math. Soc. **355** (2003), 4955-4984

MSC (2000):
Primary 14B05, 14M10, 14M25, 52B20; Secondary 13H10, 13P10, 20M25, 32S05

DOI:
https://doi.org/10.1090/S0002-9947-03-03218-5

Published electronically:
July 28, 2003

MathSciNet review:
1997591

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Based on Nakajima's Classification Theorem we describe the precise form of the binomial equations which determine toric locally complete intersection (``l.c.i.'') singularities.

**[BaMT]**M. Barile, M. Morales and A. Thoma,*Set theoretic complete intersections on binomials*, Proc. of the A.M.S.**130**, No 7, (2002), 1893-1903. MR**2003f:14058****[BiLSR]**A.-M. Bigatti, R. La Scala and L. Robbiano,*Computing toric ideals*, Jour. of Symbolic Comp.**27**(1999), 351-365. MR**2000b:13035****[BiR]**A.-M. Bigatti and L. Robbiano,*Toric ideals*, Matemática Contemporânea**21**(2001), 1-25.**[BrGT]**W. Bruns, J. Gubeladze and N.V. Trung,*Normal polytopes, triangulations and Koszul algebras*, Jour. für die reine und ang. Math.**485**(1997), 123-160. MR**99c:52016****[DHaZ]**D.I. Dais, C. Haase and G.-M. Ziegler,*All toric local complete intersection singularities admit projective crepant resolutions*, math. AG/9812025; a short version of it is published in Tôhoku Math. Journal**53**(2001), 95-107. MR**2001m:14076****[DHeZ]**D.I. Dais, M. Henk and G.-M. Ziegler,*All abelian quotient c.i.-singularities admit projective crepant resolutions in all dimensions*, Advances in Math.**139**(1998), 194-239. MR**2000b:14016****[ES]**D. Eisenbud and B. Sturmfels,*Binomial ideals*, Duke Math. Jour.**84**(1996), 1-45. MR**97d:13031****[Ew]**G. Ewald,*Combinatorial Convexity and Algebraic Geometry*, Graduate Texts in Mathematics, Vol.**168**, Springer-Verlag, (1996). MR**97i:52012****[FMSh]**K.G. Fischer, W. Morris and J. Shapiro,*Affine semigroup rings that are complete intersections*, Proc. of the A.M.S.**125**(1997), 3137-3145. MR**97m:13026****[FSh]**K.G. Fischer and J. Shapiro,*Mixed matrices and binomial ideals*, Jour. of Pure and Applied Algebra**113**(1996), 39-54. MR**97h:13008****[Fu]**W. Fulton,*Introduction to Toric Varieties*, Annals of Math. Studies, Vol.**131**, Princeton University Press, (1993). MR**94g:14028****[Ho]**M. Hochster,*Rings of invariants of tori, Cohen-Macaulay rings generated by monomials, and polytopes*, Annals of Math.**96**(1972), 318-337. MR**46:3511****[HSh]**S. Hosten and J. Shapiro,*Primary decomposition of lattice basis ideals*, Jour. of Symbolic Comp.**29**(2000), 625-639. MR**2001h:13012****[Ish]**M.-N. Ishida,*Torus embeddings and dualizing complexes*, Tôhoku Math. Jour.**32**(1980), 111-146. MR**81e:14005****[KKMS]**G. Kempf, F. Knudsen, D. Mumford and D. Saint-Donat,*Toroidal Embeddings I*, Lecture Notes in Mathematics, Vol.**339**, Springer-Verlag, (1973). MR**49:299****[Ku]**E. Kunz,*Introduction to Commutative Algebra and Algebraic Geometry*, Birkhäuser, (1985). MR**86e:14001****[M]**M. Mustata:*Jet schemes of locally complete intersection canonical singularities*, (with an appendix by D. Eisenbud and E. Frenkel), Inventiones Math.**145**, No 3 (2001), 397-424. MR**2002f:14005****[N]**H. Nakajima,*Affine torus embeddings which are complete intersections*, Tôhoku Math. Jour.**38**, (1986), 85-98. MR**87c:14058****[O]**T.Oda,*Convex Bodies and Algebraic Geometry. An Introduction to the theory of toric varieties*, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Bd.**15**, Springer-Verlag, (1988). MR**88m:14038****[R]**J.G. Rosales,*On presentations of semigroups of*, Semigroup Forum**55**, (1997), 152-159. MR**98h:20104****[RG]**J.G. Rosales and P.A. Garcia-Sanchez,*On complete intersection affine semigroups*, Comm. in Algebra**23**, No 14 (1995), 5395-5412. MR**96m:14068****[SSS]**G. Scheja, O. Scheja and U. Storch,*On regular sequences of binomials*, Manuscripta Math.**98**(1999) 115-132. MR**99k:13017****[Schr]**A. Schrijver,*Theory of Linear and Integer Programming*, Wiley Int. Pub., (1986). MR**88m:90090****[St1]**B. Sturmfels,*Gröbner Bases and Convex Polytopes*, University Lecture Series, Vol.**8**, A.M.S., (1996). MR**97b:13034****[St2]**B. Sturmfels,*Equations defining toric varieties*. In: Algebraic Geometry, Santa Cruz 1995, Proc. of Symp. in Pure Mathematics, Vol.**62**, Part II, A.M.S., (1997), 437-448. MR**99b:14058****[W]**K. Watanabe,*Invariant subrings which are complete intersections I. Invariant subrings of finite Abelian groups*, Nagoya Math. Jour.**77**(1980), 89-98. MR**82d:13020**

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Additional Information

**Dimitrios I. Dais**

Affiliation:
Department of Mathematics and Statistics, University of Cyprus, P.O. Box 20537, CY-1678 Nicosia, Cyprus

Address at time of publication:
Department of Mathematics, University of Crete, Knossos Avenue, GR-71409 Heraklion, Crete, Greece

Email:
ddais@ucy.ac.cy, ddais@math.uoc.gr

**Martin Henk**

Affiliation:
Technical University Otto von Guericke, Institute for Algebra and Geometry, PSF 4120, D-39016 Magdeburg, Germany

Email:
henk@math.uni-magdeburg.de

DOI:
https://doi.org/10.1090/S0002-9947-03-03218-5

Received by editor(s):
April 30, 2002

Received by editor(s) in revised form:
September 27, 2002

Published electronically:
July 28, 2003

Additional Notes:
The second author would like to thank the Mathematics Department of the University of Crete for hospitality and support during the spring term 2001, where this work was initiated.

Article copyright:
© Copyright 2003
American Mathematical Society