Kustin–Miller unprojection with complexes

Author:
Stavros Argyrios Papadakis

Journal:
J. Algebraic Geom. **13** (2004), 249-268

DOI:
https://doi.org/10.1090/S1056-3911-03-00350-3

Published electronically:
October 15, 2003

MathSciNet review:
2047698

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Abstract |
References |
Additional Information

Abstract: A main ingredient for the Kustin–Miller unprojection is the module $\operatorname {Hom}_R(I, \omega _R)$, where $R$ is a local Gorenstein ring and $I$ a codimension one ideal with $R/I$ Gorenstein. We prove a method of calculating it in a relative setting using resolutions. We give three applications. In the first we generalise a result of Catanese, Franciosi, Hulek, and Reid (*Embeddings of curves and surfaces*, Nagoya Math. J. **154** (1999), 185–220). The second and the third are about Tom and Jerry, two families of Gorenstein codimension four rings with $9 \times 16$ resolutions.

[Al]Al Altınok S., *Graded rings corresponding to polarised K3 surfaces and $\mathbb Q$-Fano 3-folds*. Univ. of Warwick Ph.D. thesis, Sept. 1998, 93+ vii pp.
[AK]AK Altman, A. and Kleiman, S., *Introduction to Grothendieck duality theory*. Lecture Notes in Mathematics, Vol. 146. Springer–Verlag, 1970
[BE]BE Buchsbaum D. and Eisenbud D., *Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension $3$*. Amer. J. Math. **99** (1977), 447–485
[BH]BH Bruns, W. and Herzog, J., *Cohen-Macaulay rings*. Revised edition, Cambridge Studies in Advanced Mathematics 39. CUP, 1998
[BrR]BrR Brown G. and Reid M., *Mory flips of Type A* (provisional title), in preparation
[BV]BV Bruns, W. and Vetter, U., *Determinantal rings*. Lecture Notes in Math. 1327. Springer, 1988
[CM]CM Corti A. and Mella M., *Birational geometry of terminal quartic 3-folds I*, in preparation
[CPR]CPR Corti A., Pukhlikov A. and Reid M., *Birationally rigid Fano hypersurfaces*, in Explicit birational geometry of 3-folds, A. Corti and M. Reid (eds.), CUP 2000, 175–258
[CFHR]CFHR Catanese, F., Franciosi, M., Hulek, K. and Reid, M., *Embeddings of curves and surfaces*. Nagoya Math. J. **154** (1999), 185–220
[FOV]FOV Flenner, H., O’Carrol, L. and Vogel, W., *Joins and intersections*. Springer Monographs in Mathematics. Springer–Verlag, 1999
[Ei]Ei Eisenbud, D., *Commutative algebra, with a view toward algebraic geometry*. Graduate Texts in Mathematics, 150. Springer–Verlag, 1995
[Har]Har Hartshorne, R., *Algebraic Geometry*. Graduate Texts in Mathematics, 52. Springer–Verlag, 1977
[KL]KL Kleppe H. and Laksov D., *The algebraic structure and deformation of Pfaffian schemes*. J. Algebra **64** (1980), 167–189
[KM]KM Kustin, A. and Miller, M., *Constructing big Gorenstein ideals from small ones*. J. Algebra **85** (1983), 303–322
[P]P Stavros Papadakis, Gorenstein rings and Kustin–Miller unprojection, Univ. of Warwick Ph.D. thesis, Aug 2001, vi + 72 pp., get from www.maths.warwick.ac. uk/ miles/doctors/Stavros
[PR]PR Papadakis, S. and Reid, M., *Kustin–Miller unprojection without complexes*, to appear J. Alg. Geom., math.AG/0011094, 18 pp.
[R1]R1 Reid, M., *Nonnormal del Pezzo surfaces*. Publ. Res. Inst. Math. Sci. **30** (1994), 695–727
[R2]R2 Reid, M., *Examples of Type IV unprojection*, math.AG/0108037, 16 pp.
[R3]Ki 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/ miles/3folds
[T]T Takagi, H., *On the classification of $\mathbb {Q}$-Fano 3-folds of Gorenstein index 2*. I, II, RIMS preprint 1305, Nov. 2000, 66 pp.

[Al]Al Altınok S., *Graded rings corresponding to polarised K3 surfaces and $\mathbb Q$-Fano 3-folds*. Univ. of Warwick Ph.D. thesis, Sept. 1998, 93+ vii pp.
[AK]AK Altman, A. and Kleiman, S., *Introduction to Grothendieck duality theory*. Lecture Notes in Mathematics, Vol. 146. Springer–Verlag, 1970
[BE]BE Buchsbaum D. and Eisenbud D., *Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension $3$*. Amer. J. Math. **99** (1977), 447–485
[BH]BH Bruns, W. and Herzog, J., *Cohen-Macaulay rings*. Revised edition, Cambridge Studies in Advanced Mathematics 39. CUP, 1998
[BrR]BrR Brown G. and Reid M., *Mory flips of Type A* (provisional title), in preparation
[BV]BV Bruns, W. and Vetter, U., *Determinantal rings*. Lecture Notes in Math. 1327. Springer, 1988
[CM]CM Corti A. and Mella M., *Birational geometry of terminal quartic 3-folds I*, in preparation
[CPR]CPR Corti A., Pukhlikov A. and Reid M., *Birationally rigid Fano hypersurfaces*, in Explicit birational geometry of 3-folds, A. Corti and M. Reid (eds.), CUP 2000, 175–258
[CFHR]CFHR Catanese, F., Franciosi, M., Hulek, K. and Reid, M., *Embeddings of curves and surfaces*. Nagoya Math. J. **154** (1999), 185–220
[FOV]FOV Flenner, H., O’Carrol, L. and Vogel, W., *Joins and intersections*. Springer Monographs in Mathematics. Springer–Verlag, 1999
[Ei]Ei Eisenbud, D., *Commutative algebra, with a view toward algebraic geometry*. Graduate Texts in Mathematics, 150. Springer–Verlag, 1995
[Har]Har Hartshorne, R., *Algebraic Geometry*. Graduate Texts in Mathematics, 52. Springer–Verlag, 1977
[KL]KL Kleppe H. and Laksov D., *The algebraic structure and deformation of Pfaffian schemes*. J. Algebra **64** (1980), 167–189
[KM]KM Kustin, A. and Miller, M., *Constructing big Gorenstein ideals from small ones*. J. Algebra **85** (1983), 303–322
[P]P Stavros Papadakis, Gorenstein rings and Kustin–Miller unprojection, Univ. of Warwick Ph.D. thesis, Aug 2001, vi + 72 pp., get from www.maths.warwick.ac. uk/ miles/doctors/Stavros
[PR]PR Papadakis, S. and Reid, M., *Kustin–Miller unprojection without complexes*, to appear J. Alg. Geom., math.AG/0011094, 18 pp.
[R1]R1 Reid, M., *Nonnormal del Pezzo surfaces*. Publ. Res. Inst. Math. Sci. **30** (1994), 695–727
[R2]R2 Reid, M., *Examples of Type IV unprojection*, math.AG/0108037, 16 pp.
[R3]Ki 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/ miles/3folds
[T]T Takagi, H., *On the classification of $\mathbb {Q}$-Fano 3-folds of Gorenstein index 2*. I, II, RIMS preprint 1305, Nov. 2000, 66 pp.

Additional Information

**Stavros Argyrios Papadakis**

Affiliation:
Math Institute, University of Warwick, Coventry CV4 7AL, England

Address at time of publication:
Fakultät für Mathematik und Informatik, Geb. 27, Universität des Saarlandes, D-66123 Saarbrücken, Gernamy

Email:
spapad@maths.warwick.ac.uk, papadakis@math.uni-sb.de

Received by editor(s):
August 24, 2001

Published electronically:
October 15, 2003

Additional Notes:
This work is part of a Warwick Ph.D. thesis [Gorenstein rings and Kustin–Miller unprojection, Univ. of Warwick Ph.D. thesis, Aug 2001], financially supported by the Greek State Scholarships Foundation