Let $k$ be an algebraically closed field and let $\Hilb_{d}^{aG}(\p{d-2})$ be the open locus inside the Hilbert scheme $\Hilb_{d}(\p{d-2})$ corresponding to arithmetically Gorenstein subschemes. We prove the irreducibility and characterize the singularities of $\Hilb_{6}^{aG}(\p{4})$. In order to achieve these results we also classify all Artinian, Gorenstein, not necessarily graded, $k$--algebras up to degree $6$. Moreover we describe the loci in $\Hilb_{6}^{aG}(\p{4})$ obtained via some geometric construction. Finally we prove the obstructedness of some families of points in $\Hilb_{d}^{aG}(\p{d-2})$ for each $d\ge6$.
On some Gorenstein loci in Hilb_6(P^4_k) / Casnati, Gianfranco; Notari, Roberto. - In: JOURNAL OF ALGEBRA. - ISSN 0021-8693. - 308:(2007), pp. 493-523. [10.1016/j.jalgebra.2006.09.023]
On some Gorenstein loci in Hilb_6(P^4_k)
CASNATI, GIANFRANCO;NOTARI, ROBERTO
2007
Abstract
Let $k$ be an algebraically closed field and let $\Hilb_{d}^{aG}(\p{d-2})$ be the open locus inside the Hilbert scheme $\Hilb_{d}(\p{d-2})$ corresponding to arithmetically Gorenstein subschemes. We prove the irreducibility and characterize the singularities of $\Hilb_{6}^{aG}(\p{4})$. In order to achieve these results we also classify all Artinian, Gorenstein, not necessarily graded, $k$--algebras up to degree $6$. Moreover we describe the loci in $\Hilb_{6}^{aG}(\p{4})$ obtained via some geometric construction. Finally we prove the obstructedness of some families of points in $\Hilb_{d}^{aG}(\p{d-2})$ for each $d\ge6$.Pubblicazioni consigliate
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https://hdl.handle.net/11583/1503771
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