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Reduced arithmetically Gorenstein schemes and simplicial polytopes with maximal Betti numbers. (English) Zbl 1053.13006

An SI-sequence is a finite sequence of positive integers which is symmetric, unimodal and satisfies a growth condition. From an arbitrary SI-sequence the authors construct a reduced, arithmetically Gorenstein configuration of linear varieties, \(G\), of arbitrary dimension whose Artinian reduction has the given SI-sequence as Hilbert function and has the weak Lefschetz property. It is shown that \(G\) has maximal graded Betti numbers among all arithmetically Gorenstein subschemes of the projective space whose Artinian reduction has the weak Lefschetz property and the given Hilbert function. Finally, it is shown that over a field of characteristic zero every set of simplicial polytopes with fixed \(h\)-vector contains a polytope with maximal graded Betti numbers.

MSC:

13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
13C40 Linkage, complete intersections and determinantal ideals
13D02 Syzygies, resolutions, complexes and commutative rings
14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
14M06 Linkage
14N20 Configurations and arrangements of linear subspaces
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
52B12 Special polytopes (linear programming, centrally symmetric, etc.)
52B11 \(n\)-dimensional polytopes

Software:

Macaulay2
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Full Text: DOI arXiv

References:

[1] Ballico, E.; Bolondi, G.; Ellia, P.; Mirò-Roig, R. M., Curves of maximum genus in range A and stick-figures, Trans. Amer. Math. Soc., 349, 11, 4589-4608 (1997) · Zbl 0885.14016
[2] D. Bayer, M. Stillman, Macaulay: A system for computation in algebraic geometry and commutative algebra. Source and object code available for Unix and Macintosh computers. Contact the authors, or download from ftp://math.harvard.eduviaanonymousftp; D. Bayer, M. Stillman, Macaulay: A system for computation in algebraic geometry and commutative algebra. Source and object code available for Unix and Macintosh computers. Contact the authors, or download from ftp://math.harvard.eduviaanonymousftp
[3] Bigatti, A., Upper bounds for the Betti numbers of a given Hilbert function, Comm. Algebra, 21, 7, 2317-2334 (1993) · Zbl 0817.13007
[4] Billera, L. J.; Lee, C. W., Sufficiency of McMullen’s condition for \(f\)-vectors of simplicial polytopes, Bull. Amer. Math. Soc., 2, 181-185 (1980) · Zbl 0431.52009
[5] Billera, L. J.; Lee, C. W., A proof of the sufficiency of McMullen’s conditions for \(f\)-vectors of simplicial convex polytopes, J. Combin. Theory Ser. A, 31, 237-255 (1981) · Zbl 0479.52006
[6] Björner, A.; Frankl, P.; Stanley, R., The number of faces of balanced Cohen-Macaulay complexes and a generalized Macaulay theorem, Combinatorica, 7, 23-34 (1987) · Zbl 0651.05010
[7] M. Boij, Simplicial complexes and points in projective space, unpublished.; M. Boij, Simplicial complexes and points in projective space, unpublished. · Zbl 0940.13008
[8] Boij, M., Betti numbers of compressed algebras, J. Pure Appl. Algebra, 134, 111-131 (1999) · Zbl 0941.13013
[9] Bolondi, G.; Migliore, J., The structure of an even Liaison class, Trans. Amer. Math. Soc., 316, 1-37 (1989) · Zbl 0689.14019
[10] G. Bolondi, J. Migliore, Configurations of linear projective subvarieties, in: Algebraic Curves and Projective Geometry, Proceedings (Trento, 1988), Lecture Notes in Mathematics, Vol. 1389, Springer, Berlin, 1989, pp. 19-31.; G. Bolondi, J. Migliore, Configurations of linear projective subvarieties, in: Algebraic Curves and Projective Geometry, Proceedings (Trento, 1988), Lecture Notes in Mathematics, Vol. 1389, Springer, Berlin, 1989, pp. 19-31.
[11] W. Bruns, J. Herzog, Cohen-Macaulay Rings, Revised edition, Cambridge Studies in Advanced Mathematics, Vol. 39, Cambridge University Press, Cambridge, 1998.; W. Bruns, J. Herzog, Cohen-Macaulay Rings, Revised edition, Cambridge Studies in Advanced Mathematics, Vol. 39, Cambridge University Press, Cambridge, 1998.
[12] Buchsbaum, D.; Eisenbud, D., Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension 3, Amer. J. Math, 99, 447-485 (1977) · Zbl 0373.13006
[13] Davis, E.; Geramita, A. V.; Orecchia, F., Gorenstein algebras and the Cayley-Bacharach theorem, Proc. Amer. Math. Soc., 93, 593-597 (1985) · Zbl 0575.14040
[14] Diesel, S., Irreducibility and dimension theorems for families of height 3 Gorenstein algebras, Pacific J. Math., 172, 2, 365-397 (1996) · Zbl 0882.13021
[15] Eisenbud, D.; Goto, S., Linear free resolutions and minimal multiplicity, J. Algebra, 88, 89-133 (1984) · Zbl 0531.13015
[16] Eliahou, S.; Kervaire, M., Minimal resolutions of some monomial ideals, J. Algebra, 129, 1-25 (1990) · Zbl 0701.13006
[17] R. Fröberg, D. Laksov, Compressed Algebras, in: S. Greco, R. Strano (Eds.), Complete intersections (Acireale, 1983), Lecture Notes in Mathematics, Vol. 1092, Springer, Berlin, 1984, pp. 121-151.; R. Fröberg, D. Laksov, Compressed Algebras, in: S. Greco, R. Strano (Eds.), Complete intersections (Acireale, 1983), Lecture Notes in Mathematics, Vol. 1092, Springer, Berlin, 1984, pp. 121-151.
[18] Geramita, A. V.; Gregory, D.; Roberts, L., Monomial ideals and points in projective space, J. Pure Appl. Algebra, 40, 33-62 (1986) · Zbl 0586.13015
[19] Geramita, A. V.; Harima, T.; Shin, Y. S., Extremal point sets and Gorenstein ideals, Adv. Math., 152, 78-119 (2000) · Zbl 0965.13011
[20] Geramita, A. V.; Harima, T.; Shin, Y. S., An alternative to the Hilbert function for the ideal of a finite set of points in
((P^n\), Illinois J. Math., 45, 1-23 (2001) · Zbl 1095.13500
[21] Geramita, A. V.; Ko, H. J.; Shin, Y. S., The Hilbert function and the minimal free resolution of some Gorenstein ideals of codimension 4, Comm. Algebra, 26, 4285-4307 (1998) · Zbl 0924.13016
[22] Geramita, A. V.; Kreuzer, M.; Robbiano, L., Cayley-Bacharach schemes and their canonical modules, Trans. Amer. Math. Soc., 339, 163-189 (1993) · Zbl 0793.14002
[23] Geramita, A. V.; Maroscia, P.; Roberts, L., The Hilbert function of a reduced \(k\)-algebra, J. London Math. Soc., 28, 443-452 (1983) · Zbl 0535.13012
[24] Geramita, A. V.; Migliore, J., A generalized Liaison addition, J. Algebra, 163, 139-164 (1994) · Zbl 0798.14026
[25] Geramita, A. V.; Migliore, J., Reduced Gorenstein codimension three subschemes of projective space, Proc. Amer. Math. Soc., 125, 943-950 (1997) · Zbl 0861.14040
[26] Geramita, A. V.; Pucci, M.; Shin, Y. S., Smooth points of \(Gor(T)\), J. Pure Appl. Algebra, 122, 209-241 (1997) · Zbl 0905.13004
[27] Geramita, A. V.; Shin, Y. S., \(k\)-Configurations in
((P^3\) all have extremal resolutions, J. Algebra, 213, 351-368 (1999) · Zbl 0948.13010
[28] Harima, T., Some examples of unimodal Gorenstein sequences, J. Pure Appl. Algebra, 103, 313-324 (1995) · Zbl 0847.13003
[29] Harima, T., Characterization of Hilbert functions of Gorenstein Artin algebras with the weak Stanley property, Proc. Amer. Math. Soc., 123, 3631-3638 (1995) · Zbl 0857.13013
[30] Hartshorne, R., Connectedness of the Hilbert scheme, Math. Inst. des Hautes Etudes Sci., 29, 261-304 (1966) · Zbl 0171.41502
[31] Hartshorne, R., Families of curves in
((P^3\) and Zeuthen’s problem, Mem. Amer. Math. Soc., 130, 617 (1997) · Zbl 0894.14001
[32] J. Herzog, E. Li Marzi, Bounds for the Betti numbers of shellable simplicial complexes and polytopes, Commutative Algebra and Algebraic Geometry (Ferrara), 157-167, Lecture Notes in Pure and Applied Mathematics, Vol. 206, Dekker, New York, 1999.; J. Herzog, E. Li Marzi, Bounds for the Betti numbers of shellable simplicial complexes and polytopes, Commutative Algebra and Algebraic Geometry (Ferrara), 157-167, Lecture Notes in Pure and Applied Mathematics, Vol. 206, Dekker, New York, 1999. · Zbl 0941.52013
[33] Hibi, T., Algebraic Combinatorics on Convex Polytopes (1992), Carslaw: Carslaw Glebe, Australia · Zbl 0772.52008
[34] Hibi, T.; Terai, N., Computation of Betti numbers of monomial ideals associated with stacked polytopes, Manuscripta Math., 92, 447-453 (1997) · Zbl 0882.13018
[35] Hulett, H., Maximum Betti numbers of homogeneous ideals with a given Hilbert function, Comm. Algebra, 21, 7, 2335-2350 (1993) · Zbl 0817.13006
[36] Iarrobino, A., Compressed algebrasArtin algebras having given socle degrees and maximal length, Trans. Amer. Math. Soc., 285, 337-378 (1984) · Zbl 0548.13009
[37] Iarrobino, A.; Kanev, V., Power Sums, Gorenstein Algebras, and Determinantal Loci, Lecture Notes in Mathematics, Vol. 1721 (1999), Springer: Springer Berlin · Zbl 0942.14026
[38] Ikeda, H., Results on Dilworth and Rees numbers of Artinian local rings, Japan. J. Math. (N.S.), 22, 147-158 (1996) · Zbl 0857.13014
[39] G. Kalai, Some aspects of the combinatorial theory of convex polytopes, in: Polytopes: Abstract, Convex and Computational (Scarborough, ON, 1993), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 440, Kluwer Acad. Publ., Dordrecht, 1994, pp. 205-229.; G. Kalai, Some aspects of the combinatorial theory of convex polytopes, in: Polytopes: Abstract, Convex and Computational (Scarborough, ON, 1993), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 440, Kluwer Acad. Publ., Dordrecht, 1994, pp. 205-229. · Zbl 0804.52006
[40] Kleppe, J.; Migliore, J.; Miró-Roig, R. M.; Nagel, U.; Peterson, C., Gorenstein Liaison, complete intersection Liaison invariants and unobstructedness, Mem. Amer. Math. Soc., 154, 732 (2001) · Zbl 1006.14018
[41] R. Lazarsfeld, P. Rao, Linkage of general curves of large degree, in: Algebraic Geometry-Open Problems (Ravello, 1982), Lecture Notes in Mathematics, Vol. 997, Springer, Berlin, 1983, pp. 267-289.; R. Lazarsfeld, P. Rao, Linkage of general curves of large degree, in: Algebraic Geometry-Open Problems (Ravello, 1982), Lecture Notes in Mathematics, Vol. 997, Springer, Berlin, 1983, pp. 267-289.
[42] Macaulay, F. S., Some properties of enumeration in the theory of modular systems, Proc. London Math. Soc., 26, 2, 531-555 (1927) · JFM 53.0104.01
[43] McMullen, P., The number of faces of simplicial polytopes, Israel J. Math., 9, 559-570 (1971) · Zbl 0209.53701
[44] McMullen, P., The maximum number of faces of a convex polytope, Mathematika, 17, 179-184 (1970) · Zbl 0217.46703
[45] J. Migliore, Introduction to Liaison Theory and Deficiency Modules, Progress in Mathematics, Vol. 165, Birkhäuser, Basel, 1998.; J. Migliore, Introduction to Liaison Theory and Deficiency Modules, Progress in Mathematics, Vol. 165, Birkhäuser, Basel, 1998. · Zbl 0921.14033
[46] J. Migliore, U. Nagel, Lifting monomial ideals, Comm. Algebra 28 (2000) (special volume in honour of R. Hartshorne) 5679-5701.; J. Migliore, U. Nagel, Lifting monomial ideals, Comm. Algebra 28 (2000) (special volume in honour of R. Hartshorne) 5679-5701. · Zbl 1003.13005
[47] Migliore, J.; Nagel, U., Monomial ideals and the Gorenstein liaison class of a complete intersection, Compositio Math., 133, 25-36 (2002) · Zbl 1047.14034
[48] Nagel, U., Even Liaison classes generated by Gorenstein linkage, J. Algebra, 209, 543-584 (1998) · Zbl 0933.14030
[49] Nagel, U., Arithmetically Buchsbaum divisors on varieties of minimal degree, Trans. Amer. Math. Soc., 351, 4381-4409 (1999) · Zbl 0941.14016
[50] Peskine, C.; Szpiro, L., Liaison des variétés algébriques. I, Invent. Math., 26, 271-302 (1974) · Zbl 0298.14022
[51] Reisner, G., Cohen-Macaulay quotients of polynomial rings, Adv. Math., 21, 30-49 (1976) · Zbl 0345.13017
[52] A. Ragusa, G. Zappalà, Partial intersections and graded Betti numbers, preprint, 2000.; A. Ragusa, G. Zappalà, Partial intersections and graded Betti numbers, preprint, 2000. · Zbl 1033.13004
[53] R. Stanley, Cohen-Macaulay complexes, in: Higher Combinatorics (Proceedings of NATO Advanced Study Inst., Berlin, 1976), NATO Adv. Study Inst. Ser., Ser. C: Math. and Phys. Sci., Vol. 31, Reidel, Dordrecht, 1977, pp. 51-62.; R. Stanley, Cohen-Macaulay complexes, in: Higher Combinatorics (Proceedings of NATO Advanced Study Inst., Berlin, 1976), NATO Adv. Study Inst. Ser., Ser. C: Math. and Phys. Sci., Vol. 31, Reidel, Dordrecht, 1977, pp. 51-62.
[54] R. Stanley, Combinatorics and Commutative Algebra, 2nd Edition, Progress in Mathematics, Vol. 41, Birkhäuser, Basel, 1996.; R. Stanley, Combinatorics and Commutative Algebra, 2nd Edition, Progress in Mathematics, Vol. 41, Birkhäuser, Basel, 1996. · Zbl 0838.13008
[55] Stanley, R., Hilbert functions of graded algebras, Adv. Math., 28, 57-82 (1978) · Zbl 0384.13012
[56] Stanley, R., The number of faces of a simplicial convex polytope, Adv. Math., 35, 236-238 (1980) · Zbl 0427.52006
[57] Stückrad, J.; Vogel, W., Buchsbaum Rings and Applications, An Interaction Between Algebra, Geometry and topology (1986), Springer: Springer Berlin · Zbl 0606.13017
[58] Watanabe, J., The Dilworth number of Artinian rings and finite posets with rank function, Adv. Stud. Pure Math., 11, 303-312 (1987)
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