Abstract
Let X be an irreducible and reduced excellent noetherian scheme.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
S.S. Abhyankar, Local uniformization on algebraic surfaces over ground fields of characteristic p≠0. Ann. Math. (2) 63, 491–526 (1956)
S.S. Abhyankhar, Resolution of singularities of arithmetical surfaces, in Arithmetical Algebraic Geometry (Proceedings of the Conference on Purdue University, 1963) (Harper and Row, New York, 1965), pp. 111–152
S.S. Abhyankar, An algorithm on polynomials in one indeterminate with coefficients in a two dimensional regular local domain. Ann. Mat. Pura Appl. (4) 71, 25–59 (1966)
S.S. Abhyankar, Resolution of singularities of embedded algebraic surfaces, in Pure and Applied Mathematics, vol. 24 (Academic, New York 1966), ix+291 pp.
S.S. Abhyankar, Nonsplitting of valuations in extensions of two dimensional regular local domains. Math. Ann. 170, 87–144 (1967)
S.S. Abhyankar, Good points of a hypersurface. Adv. Math. 68(2), 87–256 (1988)
G. Albanese, Trasformazione birazionale di una superficie algebrica in un’altra priva di punti multipli. Rend. Circ. Mat. Palermo 48(3), 321–332 (1924)
J.M. Aroca, H. Hironaka, J.L.J. Vicente, The Theory of the Maximal Contact. Memorias de Matemática del Instituto “Jorge Juan”, No. 29 (Consejo Superior de Investigaciones Científicas, Madrid, 1975), 135 pp.
B. Bennett, On the characteristic functions of a local ring. Ann. Math. 91, 25–87 (1970)
E. Bierstone, P.D. Milman, Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant. Invent. Math. 128(2), 207–302 (1997). (English Summary)
V. Cossart, Desingularization of embedded excellent surfaces. Tohoku Math. J. 33, 25–33 (1981)
V. Cossart, Desingularization: a few bad examples in dim. 3, characteristic p > 0, (English summary), in Topics in Algebraic and Noncommutative Geometry (Luminy/Annapolis, MD, 2001), Contemporary Mathematics, vol. 324 (American Mathematical Society, Providence, 2003), pp. 103–108
V. Cossart, O. Piltant, Resolution of singularities of threefolds in positive characteristic. I. Reduction to local uniformization on Artin-Schreier and purely inseparable coverings. J. Algebra 320(3), 1051–1082 (2008). (English Summary)
V. Cossart, O. Piltant, Resolution of singularities of threefolds in positive characteristic. II. J. Algebra 321(7), 1836–1976 (2009)
V. Cossart, O. Piltant, Resolution of singularities of arithmetical threefolds. J. Algebra 529, 268–535 (2019)
V. Cossart, B. Schober, A strictly decreasing invariant for resolution of singularities in dimension two. Publ. Res. Inst. Math. Sci. 56(2), 217–280 (2020)
V. Cossart, O. Piltant, B. Schober, Faîte du cône tangent à une singularité : un théorème oublié. Compt. Rendus Math. 355(4), 455–459 (2017)
S.D. Cutkosky, Resolution of Singularities. Graduate Studies in Mathematics, vol. 63 (American Mathematical Society, Providence, 2004), viii+186 pp.
S.D. Cutkosky, Resolution of singularities for 3-folds in positive characteristic. Amer. J. Math. 131(1), 59–127 (2009)
A.J. de Jong, Smoothness, semi-stability and alterations. Inst. Hautes Études Sci. Publ. Math. No. 83, 51–93 (1996)
S. Encinas, H. Hauser, Strong resolution of singularities in characteristic zero. Comment. Math. Helv. 77(4), 821–845 (2002). (English summary)
J. Giraud, Étude Locale des Singularités. Cours de 3e cycle, Orsay, Publication No. 26 (1971–1972). http://portail.mathdoc.fr/PMO/PDF/G_GIRAUD-50.pdf
J. Giraud, Sur la théorie du contact maximal. Math. Z. 137, 285–310 (1974). (French)
J. Giraud, Contact maximal en caractéristique positive. Ann. Sci. École Norm. Sup. (4) 8(2), 201–234 (1975). (French)
A. Grothendieck, Travaux de Heisuke Hironaka sur la résolution des singularités. Actes C.I.M. (Nice, 1970), vol. I (Gauthier-Villars, Paris, 1971), pp. 7–9
A. Grothendieck, J. Dieudonné, Éléments de géométrie algébrique IV. Publ. Math. I.H.É.S. 1: No. 20 (1964), 2: No. 24 (1965), 3: No. 28 (1966), 4: No. 32 (1967).
R. Hartshorne, Algebraic Geometry. Graduate Texts in Mathematics (Springer, Berlin, 1977)
H. Hauser, Excellent surfaces and their taut resolution, in Resolution of singularities (Obergurgl, 1997). Progress in Mathematics, vol. 181 (Birkhhäuser, Basel, 2000), pp. 341–373
H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero: I–II. Ann. Math. 79, 109–326 (1964)
H. Hironaka, On the characters ν ∗ and τ ∗ of singularities. J. Math. Kyoto Univ. 7, 19–43 (1967)
H. Hironaka, Characteristic polyhedra of singularities. J. Math. Kyoto Univ. 7, 251–293 (1967)
H. Hironaka, Desingularization of excellent surfaces, in Advanced Science Seminar in Algebraic Geometry, (Summer 1967 at Bowdoin College), Mimeographed notes by B. Bennet, Lecture Notes in Mathematics, 1101 (Springer, Berlin, 1984), pp. 99–132.
H. Hironaka, Additive groups associated with ponts of a projective space. Ann. Math. 92, 327–334 (1970)
H. Hironaka, Idealistic exponents of singularity, in Algebraic Geometry (J. J. Sylvester Symposium, Johns Hopkins University, Baltimore, 1976) (Johns Hopkins University Press, Baltimore, 1977), pp. 52–125
U. Jannsen, S. Saito, Kato conjecture and motivic cohomology over finite fields (2007, preprint, arxiv.org/abs/0910.2815)
U. Jannsen, Resolution of singularities for embedded curves, Appendix of: A finiteness theorem for zero-cycles over p-adic fields. Ann. Math. 172(3), 1593–1639 (2010).
J. Kollár, Resolution of Singularities. Annals of Mathematics Studies, vol. 166 (Princeton University Press, Princeton, 2007)
J. Lipman, Desingularization of two-dimensional schemes. Ann. Math. (2) 107(1), 151–207 (1978)
R. Narasimhan, Hyperplanarity of the equimultiple locus. Proc. Amer. Math. Soc. 87(3), 403–408 (1983)
R. Narasimhan, Monomial equimultiple curves in positive characteristic. Proc. Amer. Math. Soc. 89(3), 402–406 (1983)
V. Puiseux, Recherches sur les fonctions algébriques. J. de Math. Pures et Appl. 15, 207 (1850)
H. Reitberger The turbulent fifties in resolution of singularities, in Resolution of Singularities, ed. by H. Hauser, J. Lipman, F. Oort, A. Quirós. Progress in Mathematics, vol 181 (Birkhäuser, Basel, 2000)
J-P. Serre, Algèbre Locale Multiplicitès. Cours au Collège de France, 1957–1958, rédigé par Pierre Gabriel. Lecture Notes in Mathematics, vol. 11, 2nd edn. (Springer, Berlin, 1965), vii+188 pp.
B. Singh, Effect of a permissible blowing-up on the local Hilbert functions. Invent. Math. 26, 201–212 (1974)
M. Temkin, Desingularization of quasi-excellent schemes of characteristic zero. Adv. Math. 219, 488–522 (2008)
M. Temkin, Functorial desingularization of quasi-excellent schemes in characteristic zero: the non-embedded case. Duke J. Math. 161, 2208–2254 (2012)
U.Villamayor, Patching local uniformizations. Ann. Sci. École Norm. Sup. (4) 25(6), 629–677 (1992)
J. Walker, Resolution of singularities of an algebraic surface. Ann. Math. (2) 36, 336-365 (1935)
O. Zariski, The reduction of the singularities of an algebraic surface. Ann. Math. (2) 40, 639–689 (1939)
O. Zariski, A simplified proof for the resolution of singularities of an algebraic surface. Ann. Math. (2) 43, 583–593 (1942)
O. Zariski, Reduction of the singularities of algebraic three dimensional varieties. Ann. Math. (2) 45, 472–542 (1944)
Acknowledgements
We would like to thank our colleagues D. Cutkosky, O. Piltant, B. Schober, B. Teissier, O. Villamayor and some others whom we forgot, for long conversations on this topic and who encourage us to put this monograph in a suitable form to be published. Special thanks to B. Schober who sent us a long list of misprints and who helped us for many corrections, B. Tiefenbach who made the pictures and G. Moreno-Socias who helped us for the formatting and the readability of this book.
Many thanks to Dan Abramovich who encouraged us to publish this book and who, at the very end of the process, asked us 5 questions on the characteristic polyhedra. Great many thanks to Bernd Schober who accepted to write down, on short notice, the Appendix answering Dan Abramovich’s questions.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2020 The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Cossart, V., Jannsen, U., Saito, S. (2020). Introduction. In: Desingularization: Invariants and Strategy. Lecture Notes in Mathematics, vol 2270. Springer, Cham. https://doi.org/10.1007/978-3-030-52640-5_1
Download citation
DOI: https://doi.org/10.1007/978-3-030-52640-5_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-52639-9
Online ISBN: 978-3-030-52640-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)