Kustin–Miller unprojection without complexes

Authors:
Stavros Argyrios Papadakis and Miles Reid

Journal:
J. Algebraic Geom. **13** (2004), 563-577

DOI:
https://doi.org/10.1090/S1056-3911-04-00343-1

Published electronically:
January 27, 2004

MathSciNet review:
2047681

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Abstract |
References |
Additional Information

Abstract:

Gorenstein projection plays a key role in birational geometry; the typical example is the linear projection of a del Pezzo surface of degree $d$ to one of degree $d-1$, but variations on the same idea provide many of the classical and modern birational links between Fano 3-folds. The inverse operation is the Kustin–Miller unprojection theorem, which constructs “more complicated” Gorenstein rings starting from “less complicated” ones (increasing the codimension by 1). We give a clean statement and proof of their theorem, using the adjunction formula for the dualising sheaf in place of their complexes and Buchsbaum–Eisenbud exactness criterion. Our methods are scheme theoretic and work without any mention of the ambient space. They are thus not restricted to the local situation, and are well adapted to generalisations.

Section 2 contains examples, and discusses briefly the applications to graded rings and birational geometry that motivate this study.

[A]A S. Altınok, Graded rings corresponding to polarised K3 surfaces and $\mathbb {Q}$-Fano 3-folds, Univ. of Warwick Ph.D. thesis, September 1998, 93 + vii; see www.maths.warwick.ac.uk/$\!\sim$miles/doctors/Selma
[A1]A1 S. Altınok, Hilbert series and applications to graded rings, Int. J. Math. Math. Sci. 7 (2003), 397–403
[BE]BE David A. Buchsbaum and David Eisenbud, What makes a complex exact? J. Algebra 25 (1973) 259–268
[BH]BH W. Bruns and J. Herzog, Cohen–Macaulay rings, CUP 1993
[CFHR]CFHR F. Catanese, M. Franciosi, K. Hulek and M. Reid, Embeddings of curves and surfaces, Nagoya Math. J. 154 (1999) 185–220
[CM]CM A. Corti and M. Mella, Birational geometry of terminal quartic 3-folds

. I, preprint math.AG/0102096, 37 pp.

[CPR]CPR A. Corti, A. Pukhlikov and M. Reid, Birationally rigid Fano hypersurfaces, in Explicit birational geometry of 3-folds, A. Corti and M. Reid (eds.), CUP 2000, 175–258
[G]G Alexander Grothendieck, Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA$_2$). North-Holland and Masson & Cie, 1968
[KM]KM A. Kustin and M. Miller, Constructing big Gorenstein ideals from small ones, J. Algebra 85 (1983) 303–322
[KM0]KM0 A. Kustin and M. Miller, Algebra structures on minimal resolutions of Gorenstein rings of embedding codimension four, Math. Z. 173 (1980) 171–184
[KM1]KM1 A. Kustin and M. Miller, Deformation and linkage of Gorenstein algebras, Trans. Amer. Math. Soc. 284 (1984) 501–534
[M]M Hideyuki Matsumura, Commutative ring theory, Second edition, C.U.P., 1989
[P]P Stavros Papadakis, Gorenstein rings and Kustin–Miller unprojection, University of Warwick, Ph.D. thesis, August 2001, vi + 72 pp.; see www.maths.warwick.ac.uk/ $\!\sim$miles/doctors/Stavros
[P1]P1 Stavros Papadakis, Kustin–Miller unprojection *with* complexes, J. Algebraic Geometry 13 (2004), 249–268
[R]R M. Reid, Nonnormal del Pezzo surfaces, Math Proc. RIMS 30 (1994) 695–727
[R1]R1 M. Reid, Undergraduate commutative algebra, CUP 1995
[R2]R2 M. Reid, Graded rings over K3 surfaces, in abstracts of Matsumura memorial conference (Nagoya, August 1996), 11 pp.
[R3]Ki M. Reid, Graded rings and birational geometry, first draft in Proc. of algebraic geometry symposium (Kinosaki, October 2000), K. Ohno (Ed.), 1–72; see www.maths.warwick.ac.uk/$\!\sim$miles/3folds
[R4]T4 M. Reid, Examples of Type IV unprojection, preprint math.AG/0108037, 16 pp.
[WITO]WITO K-i Watanabe, T. Ishikawa, S. Tachibana and K. Otsuka, On tensor products of Gorenstein rings, J. Math. Kyoto Univ. 9 (1969) 413–423

[A]A S. Altınok, Graded rings corresponding to polarised K3 surfaces and $\mathbb {Q}$-Fano 3-folds, Univ. of Warwick Ph.D. thesis, September 1998, 93 + vii; see www.maths.warwick.ac.uk/$\!\sim$miles/doctors/Selma
[A1]A1 S. Altınok, Hilbert series and applications to graded rings, Int. J. Math. Math. Sci. 7 (2003), 397–403
[BE]BE David A. Buchsbaum and David Eisenbud, What makes a complex exact? J. Algebra 25 (1973) 259–268
[BH]BH W. Bruns and J. Herzog, Cohen–Macaulay rings, CUP 1993
[CFHR]CFHR F. Catanese, M. Franciosi, K. Hulek and M. Reid, Embeddings of curves and surfaces, Nagoya Math. J. 154 (1999) 185–220
[CM]CM A. Corti and M. Mella, Birational geometry of terminal quartic 3-folds

. I, preprint math.AG/0102096, 37 pp.

[CPR]CPR A. Corti, A. Pukhlikov and M. Reid, Birationally rigid Fano hypersurfaces, in Explicit birational geometry of 3-folds, A. Corti and M. Reid (eds.), CUP 2000, 175–258
[G]G Alexander Grothendieck, Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA$_2$). North-Holland and Masson & Cie, 1968
[KM]KM A. Kustin and M. Miller, Constructing big Gorenstein ideals from small ones, J. Algebra 85 (1983) 303–322
[KM0]KM0 A. Kustin and M. Miller, Algebra structures on minimal resolutions of Gorenstein rings of embedding codimension four, Math. Z. 173 (1980) 171–184
[KM1]KM1 A. Kustin and M. Miller, Deformation and linkage of Gorenstein algebras, Trans. Amer. Math. Soc. 284 (1984) 501–534
[M]M Hideyuki Matsumura, Commutative ring theory, Second edition, C.U.P., 1989
[P]P Stavros Papadakis, Gorenstein rings and Kustin–Miller unprojection, University of Warwick, Ph.D. thesis, August 2001, vi + 72 pp.; see www.maths.warwick.ac.uk/ $\!\sim$miles/doctors/Stavros
[P1]P1 Stavros Papadakis, Kustin–Miller unprojection *with* complexes, J. Algebraic Geometry 13 (2004), 249–268
[R]R M. Reid, Nonnormal del Pezzo surfaces, Math Proc. RIMS 30 (1994) 695–727
[R1]R1 M. Reid, Undergraduate commutative algebra, CUP 1995
[R2]R2 M. Reid, Graded rings over K3 surfaces, in abstracts of Matsumura memorial conference (Nagoya, August 1996), 11 pp.
[R3]Ki M. Reid, Graded rings and birational geometry, first draft in Proc. of algebraic geometry symposium (Kinosaki, October 2000), K. Ohno (Ed.), 1–72; see www.maths.warwick.ac.uk/$\!\sim$miles/3folds
[R4]T4 M. Reid, Examples of Type IV unprojection, preprint math.AG/0108037, 16 pp.
[WITO]WITO K-i Watanabe, T. Ishikawa, S. Tachibana and K. Otsuka, On tensor products of Gorenstein rings, J. Math. Kyoto Univ. 9 (1969) 413–423

Additional Information

**Stavros Argyrios Papadakis**

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

Address at time of publication:
Fakultaet fuer Mathematik und Informatik, Geb. 27, Universitaet des Saarlandes, D-66123, Saarbruecken, Germany

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

**Miles Reid**

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

Email:
miles@maths.warwick.ac.uk

Received by editor(s):
February 25, 2002

Published electronically:
January 27, 2004

Additional Notes:
The first author thanks the Greek State Scholarships Foundation for support. We both thank Kyoto University, RIMS for generous support and hospitality