Combinatorial aspects of sequences of point blowing ups. (English) Zbl 0829.14007
The authors study morphisms given by composition of a sequence of point blow ups of smooth \(d\)-dimensional varieties in terms of combinatorial information coming from the \(d\)-ary intersection form on divisors with exceptional support. The other combinatorial objects under consideration are the so-called weighted dual polyhedron, the weighted dual graph, and the weighted dual tree associated to such morphism. The weighted dual polyhedron and graph are appropriate generalizations of the weighted dual graph for \(d = 2\), the weights being only intersection numbers. The weighted tree is a way to represent combinatorially the sequence of the corresponding blowing ups. The main result of the paper claims the equivalence of the above objects; it follows from this result that the intersection form, the weighted dual polyhedron or the weighted dual graph determine the decomposition of the morphism as the sequence of blowing ups.
Reviewer: V.L.Popov (Moskva)
MSC:
| 14E15 | Global theory and resolution of singularities (algebro-geometric aspects) |
| 14B05 | Singularities in algebraic geometry |
References:
| [1] | [C]Campillo, A.: Algebroid curves in positive characteristic.Springer-Verlag LNM 813, (1980) · Zbl 0451.14010 |
| [2] | [CGL]Campillo, A.; Gonzalez-Springerg, G. and Lejeune-Jalabert, M.: Amas, ideaux complets et chaines toriques.C.R. Acad. Sciences de Paris. (1992) |
| [3] | [Ca]Casas-Alvero, E., Infinitely near imposed singularities and singularities of polar curves.Math. Ann. 287, 429–454 (1990) · Zbl 0675.14009 · doi:10.1007/BF01446904 |
| [4] | [Cs]Castellanos, J.: The dual graph for space curves.Springer-Verlag LNM 1462, (1991) · Zbl 0752.14029 |
| [5] | [DV]Du Val, P.: On absolute and non absolute singularities of algebraic surfaces.Istambul Univ. Fen. Fak. Mermvasi 9(A), 159–215 (1944) · Zbl 0063.07941 |
| [6] | [F]Fulton,: Intersection theory.Springer-Verlag EMG 2, (1983) · Zbl 0885.14002 |
| [7] | [Gr]Granja, A.: Coefficient fields for plane curves and equisingularity.Comm. Algebra 18, 193–208 (1990) · Zbl 0703.13020 · doi:10.1080/00927879008823907 |
| [8] | [Ho]Hoskin, M.A.: Zero-dimensional valuation ideals associated with plane curve branches.Proc. London Math. Soc. (3),6, 70–99 (1956) · Zbl 0070.16403 · doi:10.1112/plms/s3-6.1.70 |
| [9] | [LJ]Lejeune-Jalabert, M.: Linear systems with near base conditions and complete ideals in dimension two.Preprint, (1992) |
| [10] | [Li1]Lipman, J.: On complete ideals in regular local rings. Algebraic Geometry and Commutative Algebra, vol. I, in honor of Masayoshi Nagata.Kinokuniya, Tokyo, 203–231 (1988) |
| [11] | [Li2]Lipman, J.: Proximity inequalities for complete ideals in two-dimesional regular local rings.Commutative Algebra week, Mount Holyoke College, July 1992. Proceeding, to appear in Contemporary Mathematics |
| [12] | [N]Nuñez, A.: Anillos saturados de dimensión uno. Clasificación, significado geométrico y aplicaciones.Thesis, Univ. Valladolid, (1986) |
| [13] | [Z1]Zariski, O.: Polynomial ideals defined by infinitely near base points.Amer. J. Math. 60, 151–204 (1938) · JFM 64.0079.01 · doi:10.2307/2371550 |
| [14] | [Z2]Zariski, O.: Studies of equisingularity I, IIAm. J. Math. 87, 507–535and 972–1006 (1971) · Zbl 0132.41601 · doi:10.2307/2373019 |
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