Chapter 6 The main case: type A
A.7 Big example 2 § 6.4.2
The equations are given by the pentagrams appearing in Figure 6.3 plus some miss- ing equations. We use the notation defined in§6.4.2 and also define:
N4=ν4ν4+aξ 3 1ξ3η2
There are five missing equations involvingι(which we don’t include as they are not required for the calculation) and ten others.
Of these ten, there are four involvingz zξ3 =y2ν1N1+φ3ξ 2 1η zν4 =ν12N1+yξ 2 1η φ2ξ1+cν1 zξ7 =yν1N1ν4+ξ 2 1η φ2ξ3ξ1+cξ3ν1+φ1y2ξ1N1η zκ11=ν1N1N4+ξ 4 1N1η2 y2ξ3+aφ2ξ1η2−acν1η2 +ξ21ξ7ηφ2ξ1+cν1 three involvingξ1 ξ1ν4=yξ2ξ3+φ2y3ν1ξ1η3+ν1η2[φ3ν1η+dξ2] ξ1ξ7=ξ2ξ3ξ3+yν4η2[φ3ν1η+dξ2] +y4ξ1ν1η3 φ1N1η+bξ3 ξ1κ11=ξ2ξ3ξ7+y3ξ1ν1η3 N1η(yξ1ξ3+xν4) + (ax+by)ξ7 +N4η2[φ3ν1η+dξ2] two involving ξ2 ξ2ξ7 =ξ32ξ3+yξ1N1η3 φ1N1η+bξ3 ξ2κ11=ξ3ξ3ξ7+ξ1N1ν4η3 φ1N1η+bξ3 +yξ21ξ3N21η4
and one involvingν1:
ν1κ11=ν4N4+ξ 4 1ξ
2 3η2.
Bibliography
[BFZ] A. Berenstein, S. Fomin, A. Zelevinsky,Cluster algebras III: Upper bounds and double Bruhat cells.Duke Math. J. 126 (2005), no. 1, 1-52.
[B] E. Brieskorn, Rationale Singularit¨aten komplexer Fl¨achen.Invent. Math. 4 (1968), 336-358.
[BKR] G. Brown, M. Kerber, M. Reid, Fano 3-folds in codimension 4, Tom and Jerry, Part I.Compositio Mathematica, 148, 1171-1194.
[BR1] G. Brown, M. Reid, Diptych varieties, I. Proc. London Math. Soc. (2013) 107 (6): 1353-1394
[BR2] G. Brown, M. Reid,Diptych varieties, II: Polar varieties.Submitted, 24pp. Preprint version arXiv:1208.5858.
[BR3] G. Brown, M. Reid, Diptych varieties, III: Redundant generators. 10pp., preliminary draft available at http://homepages.warwick.ac.uk/staff/
G.Brown/diptych3.pdf.
[BR4] G. Brown, M. Reid, Diptych varieties, IV: Mori flips of type A. 18pp., preliminary draft available at http://homepages.warwick.ac.uk/staff/
G.Brown/diptych4.pdf.
[BE] D. Buchsbaum, D. Eisenbud, Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension 3. 39pp. American Journal of Mathematics. 99 (1977), no. 3, 447-485.
[C] S. Cutkosky,Elementary contractions of Gorenstein threefolds.Math. Ann. 280, pp. 521-525, 1980.
[CS] L. Chekhov, M. Shapiro, Teichmller spaces of Riemann surfaces with orb- ifold points of arbitrary order and cluster variables. Int. Math. Res. Not. IMRN, 2014:10 (2014), 2746-2772, arXiv:1111.3963.
[D] T. Ducat, Divisorial extractions from singular curves in smooth 3-folds, I.
Preprint, arXiv:1403.7614, 21pp.
[E1] D. Eisenbud,Homological algebra on a complete intersection, with an appli- cation to group representations. Transactions of the AMS. 260 (1980), no. 1, 35-64.
[E2] D. Eisenbud,Commutative algebra with a view towards algebraic geometry.
Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. [FZ] S. Fomin, A. Zelevinsky, Cluster algebras I: Foundations. J. Amer. Math.
Soc. 15 (2002), 497-529.
[GNW] S. Goto, K. Nishida, K.-i. Watanabe,Non-Cohen-Macaulay symbolic blow- ups for space monomial curves and counterexamples to Cowsiks question.
Proc. Amer. Math. Soc. 120 (1994), 383-392.
[GHK] M. Gross, P. Hacking, S. Keel,Mirror symmetry for log Calabi-Yau surfaces I.Preprint arXiv:1106.4977, 108 pp., to appear in Publ. Math. Inst. Hautes ´
Etudes Sci.
[GHKv1] M. Gross, P. Hacking, S. Keel,Mirror symmetry for log Calabi-Yau sur- faces I (version 1).Preprint arXiv:1106.4977v1, 144 pp.
[HTU] P. Hacking, J. Tevelev, G. Urz´ua, Flipping Surfaces. Preprint, arXiv:1310.1580, 65 pp., to appear in J. Algebraic Geom.
[KST] H. Kajiura, K. Saito, A. Takahashi,Matrix factorization and representations of quivers, II. Type ADE case.Adv. Math. 211 (2007), no. 1, 327-362.
[K1] Y. Kawamata, Divisorial contractions to 3-dimensional terminal quotient singularities.Higher-dimensional complex varieties (Trento, 1994), 241-246, de Gruyter, Berlin, 1996.
[K2] Y. Kawamata, Crepant blowing-up of 3-dimensional canonical singularities and its application to degenerations of surfaces.Ann. Math., Vol. 127, No. 1 (1988), 93-163.
[KM] J. Koll´ar, S. Mori, Classification of three-dimensional flips. Journal of the American Mathematical Society, Vol. 5, No. 3 (Jul., 1992), pp. 533-703. [KSB] J. Koll´ar, N. Shepherd-Barron,Threefolds and deformations of surface sin-
[M1] S. Mori, Flip theorem and the existence of minimal models for 3-folds. J. Amer. Math. Soc., 1 No. 1 (1988) 117-253.
[M2] S. Mori, On semistable extremal neighbourhoods. Higher dimensional bira- tional geometry (Kyoto, 1997), Adv. Stud. Pure Math., 35, Math. Soc. Japan, (2002), 157-184.
[PaR] S. Papadakis, M. Reid, Kustin-Miller unprojection without complexes, J. Algebraic Geometry 13 (2004), 563-577.
[PrR] Y. Prokhorov, M. Reid, On Q-Fano threefolds of Fano index 2. Preprint
arXiv:1203.0852.
[R1] M. Reid,Young person’s guide to canonical singularities.Algebraic geome- try, Bowdoin, 1985 (Brunswick, Maine, 1985), Proc. Sympos. Pure Math., 46, Providence, R.I.: American Mathematical Society, 345-414, 1987. [R2] M. Reid, What is a flip? Scanned notes, 53 pp., http://homepages.
warwick.ac.uk/~masda/3folds/what_flip.pdf.
[R3] M. Reid,Nonnormal del Pezzo surfaces.Math. Proc. RIMS 30 (1994) 695- 727.
[R4] M. Reid,Graded rings and birational geometry.Proc. of algebraic geometry symposium (Kinosaki, Oct 2000), K. Ohno (Ed.), 1-72.
[R5] M. Reid, The Du Val singularities An, Dn, E6, E7, E8. Notes, http://
homepages.warwick.ac.uk/~masda/surf/more/DuVal.pdf.
[S] J. Stevens, On the verbal deformation of cyclic quotient singularities. In Singularity theory and its applications, Part I (Coventry, 1988/1989), 302- 319, Lecture Notes in Math. 1462, Springer, 1991.
[Tz1] N. Tziolas,Terminal3-fold divisorial contractions of a surface to a curve I.
Compositio Mathematica 139, No. 3, 2003.
[Tz2] N. Tziolas,Families of D-minimal models and applications to3-fold diviso- rial contractions. Proceedings of the London Mathematical Society (2005) 90 (2), 345-370.
[Tz3] N. Tziolas, Three dimensional divisorial extremal neighbourhoods. Math Ann. 333, 2005.
[Tz4] N. Tziolas, 3-fold divisorial extremal neighbourhoods overcE7 andcE6 com-
pound Du Val singularities.Int. J. Math., 21, 1 (2010).
[Y] Y. Yoshino, Cohen-Macaulay modules over Cohen-Macaulay rings.London Mathematical Society Lecture Note Series, CUP, 1990.