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 ${\mathbb{P}}^3$.

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