Mather discrepancy and the arc spaces
[Discrépance de Mather et les espaces d’arc.]
Ishii, Shihoko
Annales de l'Institut Fourier, Tome 63 (2013), p. 89-111 / Harvested from Numdam

Le but de ce papier est une caractérisation de certaines singularités en fonction d’un nouvel invariant, la “discrépance de Mather”. Nous montrons également quelques résultats nous prouvant que la discrépance de Mather est un invariant raisonnable d’un point de vue de la géométrie birationnelle.

A goal of this paper is a characterization of singularities according to a new invariant, Mather discrepancy. We also show some evidences convincing us that Mather discrepancy is a reasonable invariant in a view point of birational geometry.

Publié le : 2013-01-01
DOI : https://doi.org/10.5802/aif.2756
Classification:  14E18,  14B05
Mots clés: singularités, espaces d’arc, log-discrépance minimale
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     author = {Ishii, Shihoko},
     title = {Mather discrepancy and the arc spaces},
     journal = {Annales de l'Institut Fourier},
     volume = {63},
     year = {2013},
     pages = {89-111},
     doi = {10.5802/aif.2756},
     zbl = {06177077},
     mrnumber = {3089196},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2013__63_1_89_0}
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Ishii, Shihoko. Mather discrepancy and the arc spaces. Annales de l'Institut Fourier, Tome 63 (2013) pp. 89-111. doi : 10.5802/aif.2756. http://gdmltest.u-ga.fr/item/AIF_2013__63_1_89_0/

[1] Ambro, Florin On minimal log discrepancies, Math. Res. Lett., Tome 6 (1999) no. 5-6, pp. 573-580 | Article | MR 1739216 | Zbl 0974.14007

[2] Ambro, Florin Inversion of adjunction for non-degenerate hypersurfaces, Manuscripta Math., Tome 111 (2003) no. 1, pp. 43-49 | Article | MR 1981595 | Zbl 1041.14022

[3] Denef, Jan; Loeser, François Germs of arcs on singular algebraic varieties and motivic integration, Invent. Math., Tome 135 (1999) no. 1, pp. 201-232 | Article | MR 1664700 | Zbl 0928.14004

[4] Ein, Lawrence; Lazarsfeld, Robert; Mustaţǎ, Mircea Contact loci in arc spaces, Compos. Math., Tome 140 (2004) no. 5, pp. 1229-1244 | Article | MR 2081163 | Zbl 1060.14004

[5] Ein, Lawrence; Mustaţă, Mircea Inversion of adjunction for local complete intersection varieties, Amer. J. Math., Tome 126 (2004) no. 6, pp. 1355-1365 | Article | MR 2102399 | Zbl 1087.14012

[6] Ein, Lawrence; Mustaţă, Mircea Jet schemes and singularities, Algebraic geometry—Seattle 2005. Part 2, Amer. Math. Soc., Providence, RI (Proc. Sympos. Pure Math.) Tome 80 (2009), pp. 505-546 | MR 2483946 | Zbl 1181.14019

[7] Ein, Lawrence; Mustaţă, Mircea; Yasuda, Takehiko Jet schemes, log discrepancies and inversion of adjunction, Invent. Math., Tome 153 (2003) no. 3, pp. 519-535 | Article | MR 2000468 | Zbl 1049.14008

[8] De Fernex, Tommaso; Docampo, R. Jacobian discrepancies and rational singularities (2011) (preprint)

[9] De Fernex, Tommaso; Ein, Lawrence; Ishii, Shihoko Divisorial valuations via arcs, Publ. Res. Inst. Math. Sci., Tome 44 (2008) no. 2, pp. 425-448 (math.AG/0701867) | Article | MR 2426354 | Zbl 1162.14023

[10] Grothendieck, A. Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. I, Inst. Hautes Études Sci. Publ. Math. (1964) no. 20, pp. 259 | Numdam | MR 173675 | Zbl 0135.39701

[11] Ishii, Shihoko Jet schemes, arc spaces and the Nash problem, C. R. Math. Acad. Sci. Soc. R. Can., Tome 29 (2007) no. 1, pp. 1-21 | MR 2354631 | Zbl 1162.14006

[12] Ishii, Shihoko Maximal divisorial sets in arc spaces, Algebraic geometry in East Asia—Hanoi 2005, Math. Soc. Japan, Tokyo (Adv. Stud. Pure Math.) Tome 50 (2008), pp. 237-249 | MR 2409559 | Zbl 1162.14022

[13] Ishii, Shihoko; Kollár, János The Nash problem on arc families of singularities, Duke Math. J., Tome 120 (2003) no. 3, pp. 601-620 | Article | MR 2030097 | Zbl 1052.14011

[14] Kawakita, Masayuki Inversion of adjunction on log canonicity, Invent. Math., Tome 167 (2007) no. 1, pp. 129-133 | Article | MR 2264806 | Zbl 1114.14009

[15] Kawakita, Masayuki On a comparison of minimal log discrepancies in terms of motivic integration, J. Reine Angew. Math., Tome 620 (2008), pp. 55-65 (mathAG/0608512) | MR 2427975 | Zbl 1151.14014

[16] Kolchin, E. R. Differential algebra and algebraic groups, Academic Press, New York (1973) (Pure and Applied Mathematics, Vol. 54) | MR 568864 | Zbl 0264.12102

[17] Shokurov, V. V. Problems about Fano varieties, Birational Geometry of Algebraic Varieties–Open Problems, Katata (1988), pp. 30-32