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].

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].

Reviewer: A.Gimigliano (Firenze)

##### 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 |