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] Altinok 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] Altman, A. and Kleiman, S., Introduction to Grothendieck duality theory. Lecture Notes in Mathematics, Vol. 146. Springer-Verlag, 1970
  • [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] Bruns, W. and Herzog, J., Cohen-Macaulay rings. Revised edition, Cambridge Studies in Advanced Mathematics 39. CUP, 1998
  • [BrR] Brown G. and Reid M., Mory flips of Type A (provisional title), in preparation
  • [BV] Bruns, W. and Vetter, U., Determinantal rings. Lecture Notes in Math. 1327. Springer, 1988
  • [CM] Corti A. and Mella M., Birational geometry of terminal quartic 3-folds I, in preparation
  • [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] Catanese, F., Franciosi, M., Hulek, K. and Reid, M., Embeddings of curves and surfaces. Nagoya Math. J. 154 (1999), 185-220
  • [FOV] Flenner, H., O'Carrol, L. and Vogel, W., Joins and intersections. Springer Monographs in Mathematics. Springer-Verlag, 1999
  • [Ei] Eisenbud, D., Commutative algebra, with a view toward algebraic geometry. Graduate Texts in Mathematics, 150. Springer-Verlag, 1995
  • [Har] Hartshorne, R., Algebraic Geometry. Graduate Texts in Mathematics, 52. Springer-Verlag, 1977
  • [KL] Kleppe H. and Laksov D., The algebraic structure and deformation of Pfaffian schemes. J. Algebra 64 (1980), 167-189
  • [KM] Kustin, A. and Miller, M., Constructing big Gorenstein ideals from small ones. J. Algebra 85 (1983), 303-322
  • [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] Papadakis, S. and Reid, M., Kustin-Miller unprojection without complexes, to appear J. Alg. Geom., math.AG/0011094, 18 pp.
  • [R1] Reid, M., Nonnormal del Pezzo surfaces. Publ. Res. Inst. Math. Sci. 30 (1994), 695-727
  • [R2] Reid, M., Examples of Type IV unprojection, math.AG/0108037, 16 pp.
  • [R3] Reid, M., Graded Rings and Birational Geometry, in Proc. of algebraic symposium (Kinosaki, Oct 2000), K. Ohno (Ed.) 1-72, available from www.maths.
  • [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 \cite{P}, financially supported by the Greek State Scholarships Foundation

American Mathematical Society