Abstract
Purpose of the present paper is to reveal the beauty and subtlety of resolution of singularities in the case of excellent two-dimensional schemes embedded in three-space and defined over an algebraically closed field of arbitrary characteristic. The proof of strong embedded resolution we describe here combines arguments and techniques of O. Zariski, H. Hironaka, S. Abhyankar and the author.
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References
Abhyankar, S.: Desingularization of plane curves. Proc. Symp. Pure Appl. Math. 40, Amer. Math. Soc. 1983.
Abhyankar, S.: Resolution of singularities of embedded algebraic surfaces. Acad. Press 1966.
Bennett, B.-M.: On the characteristic function of a local ring. Ann. Math. 91 (1970), 25–87.
Bierstone, E., Milman, P.: Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant. Invent. Math. 128 (1997), 207–302.
Cossart, V., Giraud, J., Orbanz, U.: Resolution of surface singularities. Springer Lecture Notes in Math. vol 1101.
Cossart, V.: Desingularization of embedded excellent surfaces. Tôhoku Math. J. 33 (1981), 25–33.
Cossart, V.: Uniformisation et désingularisation des surfaces d’après Zariski. This volume.
Encinas, S., Villamayor, O.: Good points and constructive resolution of singularities. Acta Math. 181 (1998), 109–158.
Encinas, S., Villamayor, O.: A course on constructive desingularization and equivariance. This volume. Excellent Surfaces 373
Giraud, J.: Etude locale des singularités. Cours de 3e Cycle, Orsay 1971/72. [Ha 1] Hauser, H.: Resolution techniques. Preprint 1999.
Hauser, H.: Seventeen obstacles for resolution of singularities. In: Singularities, The Brieskorn Anniversary Volume (eds: V. I. Arnold, G.-M. Greuel, J. Steen-brink), Birkhäuser 1998.
Hauser, H.: Tetrahedra, prismas and triangles. Preprint 1998.
Hironaka, H.: Desingularization of excellent surfaces. Notes by B. Bennett at the Conference on Algebraic Geometry, Bowdoin 1967. Reprinted in: Cossart, V., Giraud, J., Orbanz, U.: Resolution of surface singularities. Lecture Notes in Math. 1101, Springer 1984.
Hironaka, H.: Resolution of singularities of an algebraic variety over a field of characteristic zero. Ann. of Math. 79 (1964), 109–326.
Hironaka, H.: Idealistic exponents of singularity. In: Algebraic Geometry, the Johns Hopkins Centennial Lectures. Johns Hopkins University Press 1977.
Hironaka, H.: Additive groups associated with points of a projective space. Ann. Math. 92 (1970), 327–334.
Lê, D.T.: Les singularités sandwich. This volume.
Lipman, J.: Desingularization of 2-dimensional schemes. Ann. Math. 107 (1978), 151–207.
Moh, T.T.: On a Newton polygon approach to the uniformization of singularities in characteristic p. In: Algebraic Geometry and Singularities (eds.: A. Campillo, L. Narváez). Proc. Conf. on Singularities La Rábida. Birkhäuser 1996.
Orbanz, U.: Embedded resolution of algebraic surfaces after Abhyankar (characteristic 0). In: Cossart, V., Giraud, J., Orbanz, U.: Resolution of surface singularities. Lecture Notes in Math. 1101, Springer 1984.
Pfeifle, J.: Das Aufblasen algebraischer Varietäten und monomialer Ideale. Masters thesis, Innsbruck 1997.
Regensburger, G.: Die Auflösung von ebenen Kurvensingularitäten. Masters theses, Innsbruck 1999.
Rosenberg, J.: Blowing up nonreduced toric subschemes of An. Preprint 1998.
Spivakovsky, M.: A counterexample to the theorem of Beppo Levi in three dimensions. Invent. Math. 96 (1989), 181–183.
Spivakovsky, M.: Resolution of singularities. Preprint 1997.
Zariski, O.: Reduction of singularities of algebraic three dimensional varieties. Ann. Math. 45 (1944), 472–542.
Zariski, O.: The concept of a simple point of an abstract algebraic variety. Trans. Amer. Math. Soc. 62 (1947), 1–52.
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Hauser, H. (2000). Excellent Surfaces and Their Taut Resolution. In: Hauser, H., Lipman, J., Oort, F., Quirós, A. (eds) Resolution of Singularities. Progress in Mathematics, vol 181. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8399-3_13
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DOI: https://doi.org/10.1007/978-3-0348-8399-3_13
Publisher Name: Birkhäuser, Basel
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