Multigraded Hilbert schemes

Authors:
Mark Haiman and Bernd Sturmfels

Journal:
J. Algebraic Geom. **13** (2004), 725-769

DOI:
https://doi.org/10.1090/S1056-3911-04-00373-X

Published electronically:
March 15, 2004

MathSciNet review:
2073194

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Abstract |
References |
Additional Information

Abstract: We introduce the multigraded Hilbert scheme, which parametrizes all homogeneous ideals with fixed Hilbert function in a polynomial ring that is graded by any abelian group. Our construction is widely applicable, it provides explicit equations, and it allows us to prove a range of new results, including Bayer’s conjecture on equations defining Grothendieck’s classical Hilbert scheme and the construction of a Chow morphism for toric Hilbert schemes.

ArtZha01 M. Artin and J. J. Zhang, *Abstract Hilbert schemes*, Algebr. Represent. Theory **4** (2001), no. 4, 305–394.
BayPopStu01 David Bayer, Sorin Popescu, and Bernd Sturmfels, *Syzygies of unimodular Lawrence ideals*, J. Reine Angew. Math. **534** (2001), 169–186, arXiv:math.AG/9912247.
Bay82 David Bayer, *The division algorithm and the Hilbert scheme*, Ph.D. thesis, Harvard University, 1982.
BerWil00 Yuri Berest and George Wilson, *Automorphisms and ideals of the Weyl algebra*, Math. Ann. **318** (2000), no. 1, 127–147, arXiv:math.QA/0102190.
BilGelStu93 L. J. Billera, I. M. Gel’fand, and B. Sturmfels, *Duality and minors of secondary polyhedra*, J. Combin. Theory Ser. B **57** (1993), no. 2, 258–268.
Brion00 Michel Brion, *Group completions via Hilbert schemes*, J. Algebraic Geom. 12 (2003), 605–626.
Cox95 David A. Cox, *The homogeneous coordinate ring of a toric variety*, J. Algebraic Geom. **4** (1995), no. 1, 17–50, arXiv:alg-geom/9210008.
EisFloySchr01 David Eisenbud, Gunnar Floystad, and Frank-Olaf Schreyer, *Sheaf cohomology and free resolutions over exterior algebras*, Trans. Amer. Math. Soc. 355 (2003), 4397–4426.
EisHar00 David Eisenbud and Joe Harris, *The geometry of schemes*, Graduate Texts in Mathematics, vol. 197, Springer-Verlag, New York, 2000.
Evain01 Laurent Evain, *Irreducible components of the equivariant punctual Hilbert schemes*, Electronic preprint, arXiv:math.AG/0106218, 2001.
Evain02 ---, *Incidence relations among the Schubert cells of equivariant punctual Hilbert schemes*, Math. Z. **242** (2002), no. 4, 743–759, arXiv:math.AG/0005233.
Fog68 John Fogarty, *Algebraic families on an algebraic surface*, Amer. J. Math **90** (1968), 511–521.
Fulton94 William Fulton, *Introduction to toric varieties*, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993, The William H. Roever Lectures in Geometry.
GaPePlReTr99 S. Gastaminza, J. A. de la Peña, M. I. Platzeck, M. J. Redondo, and S. Trepode, *Finite-dimensional algebras with vanishing Hochschild cohomology*, J. Algebra **212** (1999), no. 1, 1–16.
Got78 Gerd Gotzmann, *Eine Bedingung für die Flachheit und das Hilbertpolynom eines graduierten Ringes*, Math. Z. **158** (1978), no. 1, 61–70.
Har66a Robin Hartshorne, *Connectedness of the Hilbert scheme*, Inst. Hautes Études Sci. Publ. Math. (1966), no. 29, 5–48.
HostMacStu01 Serkan Hosten, Diane Maclagan, and Bernd Sturmfels, *Supernormal vector configurations*, Journal of Algebraic Combinatorics, to appear.
IarrKanev99 Anthony Iarrobino and Vassil Kanev, *Power sums, Gorenstein algebras, and determinantal loci*, Lecture Notes in Mathematics, vol. 1721, Springer-Verlag, Berlin, 1999, Appendix C by Iarrobino and Steven L. Kleiman.
Iarrobino77 Anthony A. Iarrobino, *Punctual Hilbert schemes*, Mem. Amer. Math. Soc. **10** (1977), no. 188, viii+112.
Maclagan01 Diane Maclagan, *Antichains of monomial ideals are finite*, Proc. Amer. Math. Soc. **129** (2001), no. 6, 1609–1615 (electronic).
MacSmith03 Diane Maclagan and Gregory G. Smith, Multigraded Castelnuovo-Mumford Regularity, math.AC/0305214.
Mum82 David Mumford and John Fogarty, *Geometric invariant theory*, second ed., Springer-Verlag, Berlin, 1982.
Nak99 Hiraku Nakajima, *Lectures on Hilbert schemes of points on surfaces*, American Mathematical Society, Providence, RI, 1999.
Nakamura01 Iku Nakamura, *Hilbert schemes of abelian group orbits*, J. Algebraic Geom. **10** (2001), no. 4, 757–779.
PeeSti00 Irena Peeva and Mike Stillman, *Local equations for the toric Hilbert scheme*, Adv. in Appl. Math. **25** (2000), no. 4, 307–321.
PeeSti02 ---, *Toric Hilbert schemes*, Duke Math. J. **111** (2002), no. 3, 419–449.
SaitStuTak00 Mutsumi Saito, Bernd Sturmfels, and Nobuki Takayama, *Gröbner deformations of hypergeometric differential equations*, Algorithms and Computation in Mathematics, vol. 6, Springer-Verlag, Berlin, 2000.
Santos03 Francisco Santos, *Non-connected toric Hilbert schemes*, Electronic preprint, arXiv:math.CO/0204044, 2003.
Stur94 Bernd Sturmfels, *The geometry of $A$-graded algebras*, Electronic preprint, arXiv:alg-geom/9410032, 1994.
Sturmfels96 ---, *Gröbner bases and convex polytopes*, University Lecture Series, vol. 8, American Mathematical Society, Providence, RI, 1996.

ArtZha01 M. Artin and J. J. Zhang, *Abstract Hilbert schemes*, Algebr. Represent. Theory **4** (2001), no. 4, 305–394.
BayPopStu01 David Bayer, Sorin Popescu, and Bernd Sturmfels, *Syzygies of unimodular Lawrence ideals*, J. Reine Angew. Math. **534** (2001), 169–186, arXiv:math.AG/9912247.
Bay82 David Bayer, *The division algorithm and the Hilbert scheme*, Ph.D. thesis, Harvard University, 1982.
BerWil00 Yuri Berest and George Wilson, *Automorphisms and ideals of the Weyl algebra*, Math. Ann. **318** (2000), no. 1, 127–147, arXiv:math.QA/0102190.
BilGelStu93 L. J. Billera, I. M. Gel’fand, and B. Sturmfels, *Duality and minors of secondary polyhedra*, J. Combin. Theory Ser. B **57** (1993), no. 2, 258–268.
Brion00 Michel Brion, *Group completions via Hilbert schemes*, J. Algebraic Geom. 12 (2003), 605–626.
Cox95 David A. Cox, *The homogeneous coordinate ring of a toric variety*, J. Algebraic Geom. **4** (1995), no. 1, 17–50, arXiv:alg-geom/9210008.
EisFloySchr01 David Eisenbud, Gunnar Floystad, and Frank-Olaf Schreyer, *Sheaf cohomology and free resolutions over exterior algebras*, Trans. Amer. Math. Soc. 355 (2003), 4397–4426.
EisHar00 David Eisenbud and Joe Harris, *The geometry of schemes*, Graduate Texts in Mathematics, vol. 197, Springer-Verlag, New York, 2000.
Evain01 Laurent Evain, *Irreducible components of the equivariant punctual Hilbert schemes*, Electronic preprint, arXiv:math.AG/0106218, 2001.
Evain02 ---, *Incidence relations among the Schubert cells of equivariant punctual Hilbert schemes*, Math. Z. **242** (2002), no. 4, 743–759, arXiv:math.AG/0005233.
Fog68 John Fogarty, *Algebraic families on an algebraic surface*, Amer. J. Math **90** (1968), 511–521.
Fulton94 William Fulton, *Introduction to toric varieties*, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993, The William H. Roever Lectures in Geometry.
GaPePlReTr99 S. Gastaminza, J. A. de la Peña, M. I. Platzeck, M. J. Redondo, and S. Trepode, *Finite-dimensional algebras with vanishing Hochschild cohomology*, J. Algebra **212** (1999), no. 1, 1–16.
Got78 Gerd Gotzmann, *Eine Bedingung für die Flachheit und das Hilbertpolynom eines graduierten Ringes*, Math. Z. **158** (1978), no. 1, 61–70.
Har66a Robin Hartshorne, *Connectedness of the Hilbert scheme*, Inst. Hautes Études Sci. Publ. Math. (1966), no. 29, 5–48.
HostMacStu01 Serkan Hosten, Diane Maclagan, and Bernd Sturmfels, *Supernormal vector configurations*, Journal of Algebraic Combinatorics, to appear.
IarrKanev99 Anthony Iarrobino and Vassil Kanev, *Power sums, Gorenstein algebras, and determinantal loci*, Lecture Notes in Mathematics, vol. 1721, Springer-Verlag, Berlin, 1999, Appendix C by Iarrobino and Steven L. Kleiman.
Iarrobino77 Anthony A. Iarrobino, *Punctual Hilbert schemes*, Mem. Amer. Math. Soc. **10** (1977), no. 188, viii+112.
Maclagan01 Diane Maclagan, *Antichains of monomial ideals are finite*, Proc. Amer. Math. Soc. **129** (2001), no. 6, 1609–1615 (electronic).
MacSmith03 Diane Maclagan and Gregory G. Smith, Multigraded Castelnuovo-Mumford Regularity, math.AC/0305214.
Mum82 David Mumford and John Fogarty, *Geometric invariant theory*, second ed., Springer-Verlag, Berlin, 1982.
Nak99 Hiraku Nakajima, *Lectures on Hilbert schemes of points on surfaces*, American Mathematical Society, Providence, RI, 1999.
Nakamura01 Iku Nakamura, *Hilbert schemes of abelian group orbits*, J. Algebraic Geom. **10** (2001), no. 4, 757–779.
PeeSti00 Irena Peeva and Mike Stillman, *Local equations for the toric Hilbert scheme*, Adv. in Appl. Math. **25** (2000), no. 4, 307–321.
PeeSti02 ---, *Toric Hilbert schemes*, Duke Math. J. **111** (2002), no. 3, 419–449.
SaitStuTak00 Mutsumi Saito, Bernd Sturmfels, and Nobuki Takayama, *Gröbner deformations of hypergeometric differential equations*, Algorithms and Computation in Mathematics, vol. 6, Springer-Verlag, Berlin, 2000.
Santos03 Francisco Santos, *Non-connected toric Hilbert schemes*, Electronic preprint, arXiv:math.CO/0204044, 2003.
Stur94 Bernd Sturmfels, *The geometry of $A$-graded algebras*, Electronic preprint, arXiv:alg-geom/9410032, 1994.
Sturmfels96 ---, *Gröbner bases and convex polytopes*, University Lecture Series, vol. 8, American Mathematical Society, Providence, RI, 1996.

Additional Information

**Mark Haiman**

Affiliation:
Department of Mathematics, University of California, Berkeley, California 94720

**Bernd Sturmfels**

Affiliation:
Department of Mathematics, University of California, Berkeley, California 94720

MR Author ID:
238151

Received by editor(s):
June 10, 2002

Published electronically:
March 15, 2004

Additional Notes:
The first author’s research was supported in part by NSF grant DMS-0070772. The second author’s research was supported in part by NSF grant DMS-9970254