Reduced Gorenstein codimension three subschemes of projective space

Geramita, Anthony; Migliore, Juan

doi:10.1090/S0002-9939-97-03956-7

Reduced Gorenstein codimension three subschemes of projective space
HTML articles powered by AMS MathViewer

by Anthony V. Geramita and Juan C. Migliore PDF

Proc. Amer. Math. Soc. 125 (1997), 943-950 Request permission

Abstract:

It is known, from work of Diesel, which graded Betti numbers are possible for Artinian Gorenstein height three ideals. In this paper we show that any such set of graded Betti numbers in fact occurs for a reduced set of points in $\mathbb P^3$, a stick figure in $\mathbb P^4$, or more generally, a good linear configuration in $\mathbb P^n$. Consequently, any Gorenstein codimension three scheme specializes to such a “nice” configuration, preserving the graded Betti numbers in the process. This is the codimension three Gorenstein analog of a classical result of arithmetically Cohen-Macaulay codimension two schemes.

References

D. Bayer and M. Stillman, Macaulay, a computer system for computing in Commutative Algebra and Algebraic Geometry.
Giorgio Bolondi and Juan Carlos Migliore, Configurations of linear projective subvarieties, Algebraic curves and projective geometry (Trento, 1988) Lecture Notes in Math., vol. 1389, Springer, Berlin, 1989, pp. 19–31. MR 1023387, DOI 10.1007/BFb0085921
Giorgio Bolondi and Juan Carlos Migliore, The Lazarsfeld-Rao property on an arithmetically Gorenstein variety, Manuscripta Math. 78 (1993), no. 4, 347–368. MR 1208646, DOI 10.1007/BF02599318
Giorgio Bolondi and Rosa María Miró-Roig, Deformations of arithmetically Cohen-Macaulay subvarieties of $\mathbf P^n$, Manuscripta Math. 64 (1989), no. 2, 205–211. MR 998486, DOI 10.1007/BF01160119
David A. Buchsbaum and David Eisenbud, Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension $3$, Amer. J. Math. 99 (1977), no. 3, 447–485. MR 453723, DOI 10.2307/2373926
Mei-Chu Chang, A filtered Bertini-type theorem, J. Reine Angew. Math. 397 (1989), 214–219. MR 993224
C. Ciliberto, A. V. Geramita, and F. Orecchia, Remarks on a theorem of Hilbert-Burch, Boll. Un. Mat. Ital. B (7) 2 (1988), no. 3, 463–483 (English, with Italian summary). MR 963313
E. D. Davis, A. V. Geramita, and F. Orecchia, Gorenstein algebras and the Cayley-Bacharach theorem, Proc. Amer. Math. Soc. 93 (1985), no. 4, 593–597. MR 776185, DOI 10.1090/S0002-9939-1985-0776185-6
E. De Negri and G. Valla, The $h$-vector of a Gorenstein codimension three domain, Nagoya Math. J. 138 (1995), 113–140. MR 1339945, DOI 10.1017/S0027763000005201
S. Diesel, Irreducibility and Dimension Theorems for Families of Height $3$ Gorenstein Algebras, Pacific J. of Math. 172 (1966), 365–397.
Geir Ellingsrud, Sur le schéma de Hilbert des variétés de codimension $2$ dans $\textbf {P}^{e}$ à cône de Cohen-Macaulay, Ann. Sci. École Norm. Sup. (4) 8 (1975), no. 4, 423–431 (French). MR 393020, DOI 10.24033/asens.1297
C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
A. V. Geramita and J. C. Migliore, Hyperplane sections of a smooth curve in $\textbf {P}^3$, Comm. Algebra 17 (1989), no. 12, 3129–3164. MR 1030613, DOI 10.1080/00927878908823898
A. V. Geramita, M. Pucci and Y. S. Shin, Smooth points of $Gor(T)$, Queen’s Papers in Pure and Appl. Math., vol. 102, Queen’s Univ., Kingston, Ontario, 1996, pp. 256–297.
T. Harima, Some examples of unimodal Gorenstein sequences, J. Pure Appl. Algebra 103 (1995), 313–324.
R. Hartshorne, Families of Curves in $\mathbb P^3$ and Zeuthen’s Problem, preprint.
Jürgen Herzog, Ngô Viêt Trung, and Giuseppe Valla, On hyperplane sections of reduced irreducible varieties of low codimension, J. Math. Kyoto Univ. 34 (1994), no. 1, 47–72. MR 1263860, DOI 10.1215/kjm/1250519062
Jan O. Kleppe, Deformations of graded algebras, Math. Scand. 45 (1979), no. 2, 205–231. MR 580600, DOI 10.7146/math.scand.a-11837
R. Maggioni and A. Ragusa, The Hilbert function of generic plane sections of curves of $\textbf {P}^3$, Invent. Math. 91 (1988), no. 2, 253–258. MR 922800, DOI 10.1007/BF01389367
J. Migliore and C. Peterson, Construction of Codimension Three Gortenstein Subschemes of Projective Space, in preparation.
Rosa M. Miró-Roig, Nonobstructedness of Gorenstein subschemes of codimension $3$ in $\textbf {P}^n$, Beiträge Algebra Geom. 33 (1992), 131–138. MR 1163647
C. Peskine and L. Szpiro, Liaison des variétés algébriques. I, Invent. Math. 26 (1974), 271–302 (French). MR 364271, DOI 10.1007/BF01425554
Tim Sauer, Smoothing projectively Cohen-Macaulay space curves, Math. Ann. 272 (1985), no. 1, 83–90. MR 794092, DOI 10.1007/BF01455929
Richard P. Stanley, Hilbert functions of graded algebras, Advances in Math. 28 (1978), no. 1, 57–83. MR 485835, DOI 10.1016/0001-8708(78)90045-2
Bernd Ulrich, Sums of linked ideals, Trans. Amer. Math. Soc. 318 (1990), no. 1, 1–42. MR 964902, DOI 10.1090/S0002-9947-1990-0964902-8