Recovering the good component of the Hilbert scheme

Abstract

We give an explicit construction, for a flat map $X åS$ of algebraic spaces, of an ideal in the $n$’th symmetric product of $X$ over $S$. Blowing up this ideal is then shown to be isomorphic to the schematic closure in the Hilbert scheme of length $n$ subschemes of the locus of $n$ distinct points. This generalizes Haiman’s corresponding result for the affine complex plane. However, our construction of the ideal is very different from that of Haiman, using the formalism of divided powers rather than representation theory. In the nonflat case we obtain a similar result by replacing the $n$’th symmetric product by the $n$’th divided power product.

  • [altklei_gr_duality] Go to document A. Altman and S. Kleiman, Introduction to Grothendieck Duality Theory, New York: Springer-Verlag, 1970, vol. 146.
    @book {altklei_gr_duality, MRKEY = {0274461},
      AUTHOR = {Altman, Allen and Kleiman, Steven},
      TITLE = {Introduction to {G}rothendieck Duality Theory},
      SERIES = {Lecture Notes in Math},
      VOLUME={146},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1970},
      PAGES = {ii+185},
      MRCLASS = {14.55},
      MRNUMBER = {0274461},
      MRREVIEWER = {Y. Nakai},
      ZBLNUMBER = {0215.37201},
      DOI = {10.1007/BFb0060932},
      }
  • [artin_alg_form_moduli_I] M. Artin, "Algebraization of formal moduli. I," in Global Analysis, Tokyo: Univ. Tokyo Press, 1969, pp. 21-71.
    @incollection {artin_alg_form_moduli_I, MRKEY = {0260746},
      AUTHOR = {Artin, M.},
      TITLE = {Algebraization of formal moduli. {I}},
      BOOKTITLE = {Global {A}nalysis},
      VENUE={{P}apers in {H}onor of {K}. {K}odaira},
      PAGES = {21--71},
      PUBLISHER = {Univ. Tokyo Press},
      ADDRESS = {Tokyo},
      YEAR = {1969},
      MRCLASS = {14.05},
      MRNUMBER = {0260746},
      MRREVIEWER = {H. Kurke},
      ZBLNUMBER = {0205.50402},
      }
  • [BLRaynaud_neron] S. Bosch, W. Lütkebohmert, and M. Raynaud, Néron Models, New York: Springer-Verlag, 1990, vol. 21.
    @book {BLRaynaud_neron, MRKEY = {1045822},
      AUTHOR = {Bosch, Siegfried and L{ü}tkebohmert, Werner and Raynaud, Michel},
      TITLE = {Néron Models},
      SERIES = {Ergeb. Math. Grenzgeb.},
      VOLUME = {21},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1990},
      PAGES = {x+325},
      ISBN = {3-540-50587-3},
      MRCLASS = {14K15 (11G10 14L15)},
      MRNUMBER = {1045822},
      MRREVIEWER = {James Milne},
      ZBLNUMBER = {0705.14001},
      }
  • [bourbaki_polyetfractions] N. Bourbaki, Éléments de Mathématique. I: Les Structures Fondamentales de l’Analyse. Fascicule XI. Livre II: Algèbre. Chapitre 4: Polynomes et fractions rationnelles. Chapitre 5: Corps commutatifs, Paris: Hermann, 1959, vol. 1102.
    @book {bourbaki_polyetfractions, MRKEY = {0174550},
      AUTHOR = {Bourbaki, N.},
      TITLE = {\'{E}léments de Mathématique. {I}: {L}es Structures Fondamentales de l'Analyse. {F}ascicule {XI}. {L}ivre {II}: {A}lgèbre. {C}hapitre 4: {P}olynomes et fractions rationnelles. {C}hapitre 5: {C}orps commutatifs},
      SERIES = {Deuxième édition. Actualités Scientifiques et Industrielles},
      VOLUME={1102},
      PUBLISHER = {Hermann},
      ADDRESS = {Paris},
      YEAR = {1959},
      PAGES = {iv+222 pp. (2 inserts)},
      MRCLASS = {12.00},
      MRNUMBER = {0174550},
      MRREVIEWER = {E. R. Kolchin},
      }
  • [burch] Go to document L. Burch, "On ideals of finite homological dimension in local rings," Proc. Cambridge Philos. Soc., vol. 64, pp. 941-948, 1968.
    @article {burch, MRKEY = {0229634},
      AUTHOR = {Burch, Lindsay},
      TITLE = {On ideals of finite homological dimension in local rings},
      JOURNAL = {Proc. Cambridge Philos. Soc.},
      VOLUME = {64},
      YEAR = {1968},
      PAGES = {941--948},
      MRCLASS = {13.95},
      MRNUMBER = {0229634},
      MRREVIEWER = {L. J. Ratliff, Jr.},
      ZBLNUMBER = {0172.32302},
      DOI = {10.1017/S0305004100043620},
     }
  • [sga4_deligne_coh_supp_prop] Go to document P. Deligne, "Cohomologie à supports propres," in Théorie des Topos et Cohomologie Étale des Schémas. Tome 3, New York: Springer-Verlag, 1973, vol. 305, p. vi.
    @incollection{sga4_deligne_coh_supp_prop, MRKEY = {0354654},
      AUTHOR={Deligne, P.},
      TITLE = {Cohomologie à supports propres},
      BOOKTITLE={Théorie des Topos et Cohomologie Étale des Schémas. {T}ome 3},
      SERIES = {Lecture Notes in Math},
      VOLUME={305},
      NOTE = {S{é}minaire de G{é}om{é}trie Alg{é}brique du Bois-Marie 1963--1964 (SGA 4), Dirig{é} par M. Artin, A. Grothendieck et J. L. Verdier. Avec la collaboration de P. Deligne et B. Saint-Donat},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1973},
      PAGES = {vi+640},
      MRCLASS = {14-06},
      MRNUMBER = {0354654},
      ZBLNUMBER = {0255.14011},
      DOI = {10.1007/BFb0070723},
      }
  • [FGAexplained] Go to document B. Fantechi, L. Göttsche, L. Illusie, S. L. Kleiman, N. Nitsure, and A. Vistoli, Fundamental Algebraic Geometry, Providence, RI: Amer. Math. Soc., 2005, vol. 123.
    @book {FGAexplained, MRKEY = {2222646},
      AUTHOR = {Fantechi, Barbara and G{ö}ttsche, Lothar and Illusie, Luc and Kleiman, Steven L. and Nitsure, Nitin and Vistoli, Angelo},
      TITLE = {Fundamental Algebraic Geometry},
      SERIES = {Math. Surveys Monogr.},
      VOLUME = {123},
      NOTE = {Grothendieck's FGA explained},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2005},
      PAGES = {x+339},
      ISBN = {0-8218-3541-6},
      MRCLASS = {14-06 (14A15 14D15 14F20)},
      MRNUMBER = {2222646},
      MRREVIEWER = {Liam O'Carroll},
      ZBLNUMBER = {1085.14001},
      DOI = {10.1090/surv/123},
     }
  • [ferrand_norme] Go to document D. Ferrand, "Un foncteur norme," Bull. Soc. Math. France, vol. 126, iss. 1, pp. 1-49, 1998.
    @article {ferrand_norme, MRKEY = {1651380},
      AUTHOR = {Ferrand, Daniel},
      TITLE = {Un foncteur norme},
      JOURNAL = {Bull. Soc. Math. France},
      FJOURNAL = {Bulletin de la Société Mathématique de France},
      VOLUME = {126},
      YEAR = {1998},
      NUMBER = {1},
      PAGES = {1--49},
      ISSN = {0037-9484},
      CODEN = {BSMFAA},
      MRCLASS = {13B99 (16D90)},
      MRNUMBER = {1651380},
      MRREVIEWER = {Manuel Ojanguren},
      ZBLNUMBER = {1017.13005},
      URL = {http://smf4.emath.fr/Publications/Bulletin/126/html/smf_bull_126_1-49.html},
      }
  • [fogarty_algfam] Go to document J. Fogarty, "Algebraic families on an algebraic surface," Amer. J. Math, vol. 90, pp. 511-521, 1968.
    @article {fogarty_algfam, MRKEY = {0237496},
      AUTHOR = {Fogarty, John},
      TITLE = {Algebraic families on an algebraic surface},
      JOURNAL = {Amer. J. Math},
      FJOURNAL = {American Journal of Mathematics},
      VOLUME = {90},
      YEAR = {1968},
      PAGES = {511--521},
      ISSN = {0002-9327},
      MRCLASS = {14.20},
      MRNUMBER = {0237496},
      MRREVIEWER = {S. Lubkin},
      ZBLNUMBER = {0176.18401},
      DOI = {10.2307/2373541},
      }
  • [egaIV] Go to document A. Grothendieck, "Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. I," Inst. Hautes Études Sci. Publ. Math., vol. 20, p. 259, 1964.
    @article {egaIV, MRKEY = {0173675},
      AUTHOR = {Grothendieck, A.},
      TITLE = {\'{E}léments de géométrie algébrique. {IV}. \'{E}tude locale des schémas et des morphismes de schémas. {I}},
      JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
      FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
      VOLUME = {20},
      YEAR = {1964},
      PAGES = {259},
      ISSN = {0073-8301},
      MRCLASS = {14.05 (13.10)},
      MRNUMBER = {0173675},
      MRREVIEWER = {H. Hironaka},
      ZBLNUMBER = {0136.15901},
      DOI = {10.1007/BF02684747},
      }
  • [gr_existence] Go to document A. Grothendieck, "Techniques de construction et théorèmes d’existence en géométrie algébrique. IV. Les schémas de Hilbert," in Séminaire Bourbaki, Vol. 6, Paris: Soc. Math. France, 1961, p. exp. no. 221, 249-276.
    @incollection {gr_existence, MRKEY = {1611822},
      AUTHOR = {Grothendieck, Alexander},
      TITLE = {Techniques de construction et théorèmes d'existence en géométrie algébrique. {IV}. {L}es schémas de {H}ilbert},
      BOOKTITLE = {Séminaire {B}ourbaki, {V}ol. 6},
      PAGES = {Exp. No. 221, 249--276},
      PUBLISHER = {Soc. Math. France},
      ADDRESS = {Paris},
      YEAR = {1961},
      MRCLASS = {14C05},
      MRNUMBER = {1611822},
      ZBLNUMBER = {0236.14003},
      URL = {http://www.numdam.org/item?id=SB_1960-1961__6__249_0},
      }
  • [GLS_elementary] Go to document T. S. Gustavsen, D. Laksov, and R. M. Skjelnes, "An elementary, explicit, proof of the existence of Hilbert schemes of points," J. Pure Appl. Algebra, vol. 210, iss. 3, pp. 705-720, 2007.
    @article {GLS_elementary, MRKEY = {2324602},
      AUTHOR = {Gustavsen, T. S. and Laksov, D. and Skjelnes, R. M.},
      TITLE = {An elementary, explicit, proof of the existence of {H}ilbert schemes of points},
      JOURNAL = {J. Pure Appl. Algebra},
      FJOURNAL = {Journal of Pure and Applied Algebra},
      VOLUME = {210},
      YEAR = {2007},
      NUMBER = {3},
      PAGES = {705--720},
      ISSN = {0022-4049},
      CODEN = {JPAAA2},
      MRCLASS = {14C05 (16S50)},
      MRNUMBER = {2324602},
      MRREVIEWER = {Raquel Mallavibarrena},
      ZBLNUMBER = {1122.14004},
      DOI = {10.1016/j.jpaa.2006.11.003},
      }
  • [haiman_catalan] Go to document M. Haiman, "$t,q$-Catalan numbers and the Hilbert scheme," Discrete Math., vol. 193, iss. 1-3, pp. 201-224, 1998.
    @article {haiman_catalan, MRKEY = {1661369},
      AUTHOR = {Haiman, Mark},
      TITLE = {{$t,q$}-{C}atalan numbers and the {H}ilbert scheme},
      NOTE = {selected papers in honor of Adriano Garsia (Taormina, 1994)},
      JOURNAL = {Discrete Math.},
      FJOURNAL = {Discrete Mathematics},
      VOLUME = {193},
      YEAR = {1998},
      NUMBER = {1-3},
      PAGES = {201--224},
      ISSN = {0012-365X},
      CODEN = {DSMHA4},
      MRCLASS = {05E10 (05E05 13D40 14C05)},
      MRNUMBER = {1661369},
      MRREVIEWER = {A. A. Iarrobino, Jr.},
      ZBLNUMBER = {1061.05509},
      DOI = {10.1016/S0012-365X(98)00141-1},
      }
  • [iversen_lineardeterminants] Go to document B. Iversen, Linear Determinants with Applications to the Picard Scheme of a Family of Algebraic Curves, New York: Springer-Verlag, 1970, vol. 174.
    @book{iversen_lineardeterminants,
      author = {Iversen, Birger},
      TITLE = {Linear Determinants with Applications to the {P}icard Scheme of a Family of Algebraic Curves},
      SERIES={Lecture Notes in Math.},
      VOLUME={174},
      PUBLISHER={Springer-Verlag},
      ADDRESS={New York},
      YEAR={1970},
      MRCLASS = {14J05},
      MRNUMBER = {0292835},
      MRREVIEWER = {A. Mattuck},
      ZBLNUMBER = {0205.50802},
      DOI = {10.1007/BFb0069474},
      }
  • [knutson_alg_spaces] Go to document D. Knutson, Algebraic Spaces, New York: Springer-Verlag, 1971, vol. 203.
    @book {knutson_alg_spaces, MRKEY = {0302647},
      AUTHOR = {Knutson, Donald},
      TITLE = {Algebraic Spaces},
      SERIES = {Lecture Notes in Math.},
      VOLUME={203},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1971},
      PAGES = {vi+261},
      MRCLASS = {14A20 (13J15 14F20)},
      MRNUMBER = {0302647},
      MRREVIEWER = {H. Kurke},
      ZBLNUMBER = {0221.14001},
      DOI = {10.1007/BFb0059750},
      }
  • [kollar_rat_curves] J. Kollár, Rational Curves on Algebraic Varieties, New York: Springer-Verlag, 1996, vol. 32.
    @book {kollar_rat_curves, MRKEY = {1440180},
      AUTHOR = {Koll{á}r, J{á}nos},
      TITLE = {Rational Curves on Algebraic Varieties},
      SERIES = {Ergeb. Math. Grenzgeb.},
      VOLUME = {32},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1996},
      PAGES = {viii+320},
      ISBN = {3-540-60168-6},
      MRCLASS = {14-02 (14C05 14E05 14F17 14J45)},
      MRNUMBER = {1440180},
      MRREVIEWER = {Yuri G. Prokhorov},
      ZBLNUMBER = {0877.14012},
     }
  • [northcott] Go to document D. G. Northcott, Finite Free Resolutions, Cambridge: Cambridge Univ. Press, 1976, vol. 71.
    @book {northcott, MRKEY = {0460383},
      AUTHOR = {Northcott, D. G.},
      TITLE = {Finite Free Resolutions},
      SERIES = {Cambridge Tracts in Math},
      VOLUME={ 71},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {1976},
      PAGES = {xii+271},
      MRCLASS = {16A62 (18G10)},
      MRNUMBER = {0460383},
      MRREVIEWER = {Melvin Hochster},
      ZBLNUMBER = {0328.13010},
      DOI = {10.1017/CBO9780511565892},
     }
  • [roby_lois_pol_mod] Go to document N. Roby, "Lois polynomes et lois formelles en théorie des modules," Ann. Sci. École Norm. Sup., vol. 80, pp. 213-348, 1963.
    @article {roby_lois_pol_mod, MRKEY = {0161887},
      AUTHOR = {Roby, Norbert},
      TITLE = {Lois polynomes et lois formelles en théorie des modules},
      JOURNAL = {Ann. Sci. École Norm. Sup.},
      FJOURNAL = {Annales Scientifiques de l'École Normale Supérieure. Troisième Série},
      VOLUME = {80},
      YEAR = {1963},
      PAGES = {213--348},
      ISSN = {0012-9593},
      MRCLASS = {14.50 (13.50)},
      MRNUMBER = {0161887},
      MRREVIEWER = {P. Samuel},
      ZBLNUMBER = {0117.02302},
      URL = {http://www.numdam.org/item?id=ASENS_1963_3_80_3_213_0},
      }
  • [roby_lois_pol_mult] N. Roby, "Lois polynômes multiplicatives universelles," C. R. Acad. Sci. Paris Sér. A-B, vol. 290, iss. 19, p. a869-a871, 1980.
    @article {roby_lois_pol_mult, MRKEY = {0580160},
      AUTHOR = {Roby, Norbert},
      TITLE = {Lois polynômes multiplicatives universelles},
      JOURNAL = {C. R. Acad. Sci. Paris Sér. A-B},
      FJOURNAL = {Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences. Séries A et B},
      VOLUME = {290},
      YEAR = {1980},
      NUMBER = {19},
      PAGES = {A869--A871},
      ISSN = {0151-0509},
      CODEN = {CHASAP},
      MRCLASS = {13C99},
      MRNUMBER = {0580160},
      MRREVIEWER = {C. R. Fletcher},
      ZBLNUMBER = {0471.13008},
      }
  • [rydh_fams] D. Rydh, Families of zero cycles and divided powers: I. Representability, 2008.
    @misc{rydh_fams,
      author = {Rydh, David},
      TITLE = {Families of zero cycles and divided powers: {I.} {R}epresentability},
      ARXIV={0803.0618},
      YEAR={2008},
     }
  • [RydhHilbert] Go to document D. Rydh, "Representability of Hilbert schemes and Hilbert stacks of points," Comm. Algebra, vol. 39, iss. 7, pp. 2632-2646, 2011.
    @article {RydhHilbert, MRKEY = {2821738},
      AUTHOR = {Rydh, David},
      TITLE = {Representability of {H}ilbert schemes and {H}ilbert stacks of points},
      JOURNAL = {Comm. Algebra},
      FJOURNAL = {Communications in Algebra},
      VOLUME = {39},
      YEAR = {2011},
      NUMBER = {7},
      PAGES = {2632--2646},
      ISSN = {0092-7872},
      CODEN = {COALDM},
      MRCLASS = {14C05 (14A20 14D23)},
      MRNUMBER = {2821738},
      MRREVIEWER = {Li Li},
      ZBLNUMBER = {1235.14007},
      DOI = {10.1080/00927872.2010.488678},
      }
  • [weyl_invariants] H. Weyl, The Classical Groups. Their Invariants and Representations, Princeton, NJ: Princeton Univ. Press, 1939.
    @book{weyl_invariants,
      author={Weyl, Hermann},
      TITLE={The {C}lassical {G}roups. {T}heir {I}nvariants and {R}epresentations},
      PUBLISHER={Princeton Univ. Press},
      ADDRESS={Princeton, NJ},
      YEAR={1939},
      JFMNUMBER = {65.0058.02},
     }

Authors

Torsten Ekedahl

Roy Skjelnes

KTH, Royal Institute of Technology, Stockholm, Sweden