Abstract
We prove a formula for Thom polynomials of $A_d$ singularities in any codimension. We use a combination of the test-curve model of Porteous, and the localization methods in equivariant cohomology. Our formulas are independent of the codimension, and are computationally effective up to $d=6$.
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AUTHOR={Arnold, V. I. and Goryunov, V. V. and Lyashko, O. V. and Vasilliev, V. A.},
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ADDRESS = {New York},
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ISBN = {3-540-50583-0},
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SERIES = {Grad. Texts in Math.},
VOLUME = {227},
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ADDRESS = {New York},
YEAR = {2005},
PAGES = {xiv+417},
ISBN = {0-387-22356-8},
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@book {newstead, MRKEY = {0546290},
AUTHOR = {Newstead, P. E.},
TITLE = {Introduction to Moduli Problems and Orbit Spaces},
SERIES = {Tata Inst. Fund. Res. Lect. Math. Phys.},
VOLUME = {51},
PUBLISHER = {Tata Institute of Fundamental Research},
ADDRESS = {Bombay},
YEAR = {1978},
PAGES = {vi+183},
ISBN = {0-387-08851-2},
MRCLASS = {14-02 (14D20)},
MRNUMBER = {0546290},
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ZBLNUMBER = {0411.14003},
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[forporteous]
I. R. Porteous, "Simple singularities of maps," in Proceedings of Liverpool Singularities Symposium, I (1969/70), New York, 1971, pp. 286-307.
@inproceedings {forporteous, MRKEY = {0293646},
AUTHOR = {Porteous, Ian R.},
TITLE = {Simple singularities of maps},
BOOKTITLE = {Proceedings of {L}iverpool {S}ingularities {S}ymposium, {I} (1969/70)},
PAGES = {286--307},
SERIES={ Lecture Notes in Math.},
VOLUME={192},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1971},
MRCLASS = {57D45 (14E15)},
MRNUMBER = {0293646},
MRREVIEWER = {J. M. Boardman},
ZBLNUMBER = {0221.57016},
DOI = {10.1007/BFb0066829},
} -
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@incollection {por, MRKEY = {0713263},
AUTHOR = {Porteous, Ian R.},
TITLE = {Probing singularities},
BOOKTITLE = {Singularities, {P}art 2},
VENUE={{A}rcata, {C}alif., 1981},
SERIES = {Proc. Sympos. Pure Math.},
VOLUME = {40},
PAGES = {395--406},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, R.I.},
YEAR = {1983},
MRCLASS = {58C27 (57R45)},
MRNUMBER = {0713263},
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ZBLNUMBER = {0523.58010},
} -
[prag] P. Pragacz, Thom polynomials and Schur-functions I..
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author={Pragacz, P.},
TITLE={Thom polynomials and {S}chur-functions {I.}},
ARXIV={math.AG/0509234},
YERA={2005},
} -
[sigma11] F. Ronga, "Le calcul des classes duales aux singularites de Boardman d’ordre 2," C. R. Acad. Sci. Paris Sér. A-B, vol. 270, p. a582-a584, 1970.
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AUTHOR = {Ronga, Felice},
TITLE = {Le calcul des classes duales aux singularites de {B}oardman d'ordre 2},
JOURNAL = {C. R. Acad. Sci. Paris Sér. A-B},
VOLUME = {270},
YEAR = {1970},
PAGES = {A582--A584},
MRCLASS = {57.20},
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ZBLNUMBER = {0191.54202},
} -
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AUTHOR = {Ronga, Felice},
TITLE = {A new look at {F}aà de {B}runo's formula for higher derivatives of composite functions and the expression of some intrinsic derivatives},
BOOKTITLE = {Singularities, {P}art 2},
VENUE={{A}rcata, {C}alif., 1981},
SERIES = {Proc. Sympos. Pure Math.},
VOLUME = {40},
PAGES = {423--431},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {1983},
MRCLASS = {57R45 (26B12 58C20 58C27)},
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} -
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JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {143},
YEAR = {2001},
NUMBER = {3},
PAGES = {499--521},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {58K30 (57R45 57R70)},
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TITLE = {Equivariant multiplicities on complex varieties},
BOOKTITLE = {Orbites Unipotentes et Repr{é}sentations, {III}},
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ISSN = {0373-0956},
MRCLASS = {53.0X},
MRNUMBER = {0087149},
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ZBLCOMMENT = {BIBPROC: YEAR doesn't match found ZBLNUMBER},
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} -
[voj]
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@article {voj, MRKEY = {1072817},
AUTHOR = {Vergne, Mich{è}le},
TITLE = {Polynômes de {J}oseph et représentation de {S}pringer},
JOURNAL = {Ann. Sci. École Norm. Sup.},
FJOURNAL = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
VOLUME = {23},
YEAR = {1990},
NUMBER = {4},
PAGES = {543--562},
ISSN = {0012-9593},
CODEN = {ASENAH},
MRCLASS = {17B15 (17B35 22E60)},
MRNUMBER = {1072817},
MRREVIEWER = {Anthony Joseph},
URL = {http://www.numdam.org/item?id=ASENS_1990_4_23_4_543_0},
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}