A network of rational curves on the Hilbert scheme

Details A network of rational curves on the Hilbert scheme A network of rational curves on the Hilbert scheme

2010-06-25

arxiv.org/abs/1006.5020v2

Mathematics

Commutative Algebra

Typos corrected. Details about the degree of the curves added. References added. Associated software available at http://www

Scientific paper

In this paper we introduce an effective method to construct rational deformations between couples of Borel-fixed ideals. These deformations are governed by flat families, so that they correspond to rational curves on the Hilbert scheme. Looking globally at all the deformations among Borel-fixed ideals defining points on the same Hilbert scheme, we are able to give a new proof of the connectedness of the Hilbert scheme and to introduce a new criterion to establish whenever a set of points defined by Borel ideals lies on a common component of the Hilbert scheme. The paper contains a detailed algorithmic description of the technique and all the algorithms are made available.

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