Skip to Main Content
Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Kustin–Miller unprojection with complexes

Author: Stavros Argyrios Papadakis
Journal: J. Algebraic Geom. 13 (2004), 249-268
Published electronically: October 15, 2003
MathSciNet review: 2047698
Full-text PDF

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.

References [Enhancements On Off] (What's this?)

    [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 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. 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

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