Codimension $3$ Arithmetically Gorenstein Subschemes of projective $N$-space

Hartshorne, Robin; Sabadini, Irene; Schlesinger, Enrico

Annales de l'Institut Fourier, Tome 58 (2008) no. 6, pp. 2037-2073.

Nous étudions le problème de savoir si tous les sous-schémas arithmétiquement de Cohen-Macaulay de ${\mathbb{P}}^N$ sont “glicci” dans le cas de plus petite dimension, c’est-à-dire le cas de sous-schémas de dimension zéro de ${\mathbb{P}}^3$. Nous prouvons qu’il n’y a pas de liaisons ni de biliaisons de Gorenstein descendantes d’un ensemble d’au moins 56 points généraux de ${\mathbb{P}}^3$. Pour démontrer ce théorème, nous établissons plusieurs résultats concernant les sous-schémas arithmétiquement de Gorenstein de ${\mathbb{P}}^3$.

We study the lowest dimensional open case of the question whether every arithmetically Cohen–Macaulay subscheme of ${\mathbb{P}}^N$ is glicci, that is, whether every zero-scheme in ${\mathbb{P}}^3$ is glicci. We show that a general set of $n\ge 56$ points in ${\mathbb{P}}^3$ admits no strictly descending Gorenstein liaison or biliaison. In order to prove this theorem, we establish a number of important results about arithmetically Gorenstein zero-schemes in ${\mathbb{P}}^3$.

Classification : 14C20, 14H50, 14M06, 14M07
Keywords: Gorenstein liaison, zero-dimensional schemes, $h$-vector
Mot clés : liaison de Gorenstein, schéma de dimension zéro, vecteur $h$

[1] Boij, M. Gorenstein Artin algebras and points in projective space, Bull. London Math. Soc., Volume 31 (1997), pp. 11-16 | DOI | MR | Zbl

[2] Bruns, W.; Herzog, J. Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, Volume 39, Cambridge University Press, 1993 | MR | Zbl

[3] Buchsbaum, D. A.; Eisenbud, D. Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension 3, Amer. J. Math., Volume 99 (1977), pp. 447-485 | DOI | MR | Zbl

[4] Casanellas, M.; Drozd, E.; Hartshorne, R. Gorenstein liaison and ACM sheaves, J. Reine Angew. Math., Volume 584 (2005), pp. 149-171 | DOI | MR | Zbl

[5] Casanellas, M.; Hartshorne, R. Gorenstein biliaison and ACM sheaves, J. Algebra, Volume 278 (2004), pp. 314-341 | DOI | MR | Zbl

[6] Conca, A.; Valla, G. Hilbert function of powers of ideals of low codimension, Math. Z., Volume 230 (1999), pp. 753-784 | DOI | MR | Zbl

[7] Davis, E.; Geramita, A. V.; Orecchia, F. Gorenstein algebras and the Cayley-Bacharach Theorem, Proc. Amer. Math. Soc., Volume 93 (1985), pp. 593-597 | DOI | MR | Zbl

[8] De Negri, E.; Valla, G. The $h$-vector of a Gorenstein codimension three domain, Nagoya Math. J., Volume 138 (1995), pp. 113-140 | MR | Zbl

[9] Diesel, S. J. Irreducibility and dimension theorems for families of height 3 Gorenstein algebras, Pacific J. Math., Volume 172 (1996), pp. 365-397 | MR | Zbl

[10] Ellia, Ph. Exemples de courbes de ${\mathbb{P}}^3$ à fibré normal semi-stable, stable, Math. Ann., Volume 264 (1983), pp. 389-396 | DOI | MR | Zbl

[11] Ellia, Ph. Double structures and normal bundle of space curves, J. London Math. Soc., Volume 58 (1998), pp. 18-26 | DOI | MR | Zbl

[12] Ellingsrud, G. Sur le schéma de Hilbert des variétés de codimension 2 dans ${\mathbb{P}}^e$ à cône de Cohen–Macaulay, Ann. Sci. ENS, Volume 8 (1975), pp. 423-432 | Numdam | MR | Zbl

[13] Gruson, L.; Peskine, C. Genre des courbes de l’espace projectif, Springer LNM, Volume 687 (1977), pp. 31-59 | Zbl

[14] Hartshorne, R. Ample Subvarieties of Algebraic Varieties, Lectures Notes in Math., 156, Springer Verlag, Heidelberg, 1970 | MR | Zbl

[15] Hartshorne, R. On the classification of algebraic space curves. II, Algebraic geometry, Bowdoin, Volume 46 (1985), pp. 145-164 (Proc. Sympos. Pure Math., Part 1, Amer. Math. Soc., Providence, RI, 1987) | MR | Zbl

[16] Hartshorne, R. Generalized divisors on Gorenstein schemes, $K$-Theory, Volume 8 (1994), pp. 287-339 | DOI | MR | Zbl

[17] Hartshorne, R. Some examples of Gorenstein liaison in codimension three, Collect. Math., Volume 53 (2002), pp. 21-48 | MR | Zbl

[18] Hartshorne, R. Geometry of arithmetically Gorenstein curves in ${\mathbb{P}}^4$, Collect. Math., Volume 55 (2004) no. 1, pp. 97-111 | MR | Zbl

[19] Hartshorne, R. Generalized divisors and biliaison, Illinois J. Math., Volume 51 (2007), pp. 83-98 | MR | Zbl

[20] Herzog, J.; Trung, N. V.; Valla, G. On hyperplane sections of reduced irreducible varieties of low codimension, J. Math. Kyoto Univ., Volume 34 (1994), pp. 47-72 | MR | Zbl

[21] Kleppe, J. O. Maximal Families of Gorenstein Algebras, Trans. Amer. Math. Soc., Volume 358 (2006), pp. 3133-3167 | DOI | MR | Zbl

[22] Kleppe, J. O. Maximal Families of Gorenstein Algebras, Trans. Amer. Math. Soc., Volume 358 (2006), pp. 3133-3167 | DOI | MR | Zbl

[23] Kleppe, J. O.; Migliore, J. C.; Miró–Roig, R. M.; Nagel, U.; Peterson, C. Gorenstein liaison, complete intersection liaison invariants and unobstructedness, Memoirs Amer. Math. Soc., 2001 no. 732 | Zbl

[24] Kleppe, J. O.; Miró–Roig, R. M. The dimension of the Hilbert scheme of Gorenstein codimension 3 subschemes, J. Pure Appl. Algebra, Volume 127 (1998), pp. 73-82 | DOI | MR | Zbl

[25] Kreuzer, M. On $0$-dimensional complete intersections, Math. Ann., Volume 292 (1992), pp. 43-58 | DOI | MR | Zbl

[26] Macaulay, F. S. Some properties of enumeration in the theory of modular systems, Proc. London Math. Soc., Volume 26 (1927), pp. 531-555 | DOI

[27] Martin–Deschamps, M.; Perrin, D. Sur la classification des courbes gauches, Astérisque, 184–185, Société Mathématique de France, 1990 | Numdam | MR | Zbl

[28] Migliore, J. C. Introduction to Liaison Theory and Deficiency Modules, Birkhäuser, Boston, 1998 | MR | Zbl

[29] Migliore, J. C.; Nagel, U. Monomial ideals and the Gorenstein liaison class of a complete intersection, Compositio Math., Volume 133 (2002), pp. 25-36 | DOI | MR | Zbl

[30] Nollet, S. Bounds on multisecant lines, Collect. Math., Volume 49 (1998), pp. 447-463 | MR | Zbl

[31] Perrin, D. Courbes passant par $m$ points généraux de ${\mathbf{P}}^3$, Mémoires, 28–29, Soc. Math. France (N.S.), 1987 | Numdam | MR | Zbl

[32] Sernesi, E. Topics on families of projective schemes, Queen’s Papers in Pure and Applied Mathematics, Volume 73, Queen’s University, Kingston, 1986

[33] Stanley, R. P. Hilbert functions of graded algebras, Advances in Math., Volume 28 (1978), pp. 57-83 | DOI | MR | Zbl

[34] Walter, C. Algebraic cohomology methods for the normal bundle of algebraic space curves (1990) (Preprint)

[35] Watanabe, J. A note on Gorenstein rings of embedding codimension three, Nagoya Math. J., Volume 50 (1973), pp. 227-232 | MR | Zbl