# zbMATH — the first resource for mathematics

Resolutions of $$0$$-dimensional subschemes of a smooth quadric. (English) Zbl 0826.14029
Orecchia, Ferruccio (ed.) et al., Zero-dimensional schemes. Proceedings of the international conference held in Ravello, Italy, June 8-13, 1992. Berlin: de Gruyter. 191-204 (1994).
In this paper 0-dimensional subschemes $$X$$ of a smooth quadric $$Q \subseteq \mathbb{P}^3$$ are studied. The main problem which is treated is the following: how to recover information about $$X$$ as subscheme of $$\mathbb{P}^3$$ (i.e. information about the ideal sheaf $${\mathcal I}_X \subseteq {\mathcal O}_{\mathbb{P}^3})$$, from information on $$X$$ as a subscheme of $$Q$$ (i.e., information about the ideal sheaf $$\overline {\mathcal I}_X \subseteq {\mathcal O}_Q)$$.
The main results in the paper are about how to recover a minimal resolution of $${\mathcal I}_X$$ from a minimal resolution of $$\overline {\mathcal I}_X$$ (as $${\mathcal O}_Q$$-module): Let $$0 \to {\mathcal F}_2 \to {\mathcal F}_1 \to {\mathcal F}_0 \to \overline {\mathcal I}_X \to 0$$, be a free resolution of $$\overline {\mathcal I}_X$$, where $${\mathcal F}_i \cong \bigoplus^{n_j}_{j = 1} {\mathcal O}_Q (- a_{ij}, - a_{ij}')$$: since the resolutions of the $${\mathcal F}_i$$ as $${\mathcal O}_{\mathbb{P}^3}$$-modules are known, a double mapping cone gives a resolution of $$\overline {\mathcal I}_X$$ as $${\mathcal O}_{\mathbb{P}^3}$$-module, possibly not minimal. From such resolution on gets a resolution of $${\mathcal I}_X$$ just adding $${\mathcal O}_{\mathbb{P}^3} (- 2) \cong {\mathcal I}_Q$$ to the sheaf of generators (assuming that the quadric is a minimal generator for $${\mathcal I}_X$$, the exceptions are easy to treat). Hence the problem is how to “minimize” the resolution that one gets from the mapping cone constructions; this is done either using Horrocks’ criterion or with an “ad hoc” lemma, which works in several cases, e.g. when $${\mathcal I}_X$$ is perfect or for some kind of determinantal $$X$$, i.e. with a length 2 resolution: $$0 \to {\mathcal F}_1 \to {\mathcal F}_0 \to \overline {\mathcal I}_X \to 0$$. Eventually, some results are given on generators and syzygies of $$\overline {\mathcal I}_X$$ for points in generic position on $$Q$$.
For the entire collection see [Zbl 0797.00007].

##### MSC:
 14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) 14E15 Global theory and resolution of singularities (algebro-geometric aspects) 13D02 Syzygies, resolutions, complexes and commutative rings 14C20 Divisors, linear systems, invertible sheaves