The Iarrobino scheme: a self-dual analogue of the Hilbert scheme of points
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909760220037120 |
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| author | Jelisiejew, Joachim |
| author_facet | Jelisiejew, Joachim |
| contents | For a fixed quasi-projective scheme $X$ we introduce a self-dual analogue of ${\mathrm{Hilb}}_d(X)$ which we call the Iarrobino scheme of $X$. It is a fine moduli space of oriented Gorenstein zero-dimensional subschemes of $X$ together with some additional data (a self-dual filtration) which is vacuous over a big open set but non-trivial over the compactification. Via the link between Hilbert schemes and varieties of commuting matrices, Iarrobino schemes correspond to commuting symmetric matrices.
We provide also self-dual analogues of the Quot scheme of points and of the stacks of coherent sheaves and finite algebras. A crucial role in the construction is played by the variety of completed quadrics. We prove that the resulting analogues of Hilbert and Quot schemes are smooth for $X$ a smooth curve and that they have very rich geometry. We give applications, in particular to deformation theory of (usual) Hilbert schemes of points on threefolds, and to enumerative geometry à la June Huh. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_21705 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Iarrobino scheme: a self-dual analogue of the Hilbert scheme of points Jelisiejew, Joachim Algebraic Geometry Commutative Algebra 14C05, 14M15, 14N10, 13E10, 13H10 For a fixed quasi-projective scheme $X$ we introduce a self-dual analogue of ${\mathrm{Hilb}}_d(X)$ which we call the Iarrobino scheme of $X$. It is a fine moduli space of oriented Gorenstein zero-dimensional subschemes of $X$ together with some additional data (a self-dual filtration) which is vacuous over a big open set but non-trivial over the compactification. Via the link between Hilbert schemes and varieties of commuting matrices, Iarrobino schemes correspond to commuting symmetric matrices. We provide also self-dual analogues of the Quot scheme of points and of the stacks of coherent sheaves and finite algebras. A crucial role in the construction is played by the variety of completed quadrics. We prove that the resulting analogues of Hilbert and Quot schemes are smooth for $X$ a smooth curve and that they have very rich geometry. We give applications, in particular to deformation theory of (usual) Hilbert schemes of points on threefolds, and to enumerative geometry à la June Huh. |
| title | The Iarrobino scheme: a self-dual analogue of the Hilbert scheme of points |
| topic | Algebraic Geometry Commutative Algebra 14C05, 14M15, 14N10, 13E10, 13H10 |
| url | https://arxiv.org/abs/2508.21705 |