| 1. |
- Gustavsen, T. S., et al.
(författare)
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An elementary, explicit, proof of the existence of Hilbert schemes of points
- 2007
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Ingår i: Journal of Pure and Applied Algebra. - : Elsevier BV. - 0022-4049 .- 1873-1376. ; 210:3, s. 705-720
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Tidskriftsartikel (refereegranskat)abstract
- There are many beautiful constructions of the Hilbert scheme of closed subschemes of projective schemes. In many cases where Hilbert schemes are involved, it suffices to know they exist. On the other hand, there are many situations when it is crucial to have an explicit description of the Hilbert schemes. In this article we give a simple construction of Hilbert schemes of points, under general circumstances, that provides such a description. In addition to being useful for computations the construction is short, elementary, and explicit. Our methods rely on simple algebraic constructions. It gives explicit expressions of members of an affine covering of the Hilbert schemes, involving few variables satisfying natural equations. In particular, it explains the connections with commuting schemes of matrices. We also give some examples showing how our method can be used.
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| 2. |
- Gustavsen, Trond Stolen, et al.
(författare)
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An elementary, explicit, proof of the existence of quot schemes of points
- 2007
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Ingår i: Pacific Journal of Mathematics. - : Mathematical Sciences Publishers. - 0030-8730 .- 1945-5844. ; 231:2, s. 401-415
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Tidskriftsartikel (refereegranskat)abstract
- We give an easy and elementary construction of quotient schemes of modules of relatively finite rank under very general conditions. The construction provides a natural, explicit description of an affine covering of the quotient schemes, that is useful in many situation.
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| 3. |
- Heinrich, Katharina, et al.
(författare)
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The space of twisted cubics
- 2021
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Ingår i: Epijournal de Geometrie Algebrique. - : Centre pour la Communication Scientifique Directe (CCSD). - 2491-6765. ; 5, s. 1-22
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Tidskriftsartikel (refereegranskat)abstract
- We consider the Cohen-Macaulay compactification of the space of twisted cubics in Pn. This compactification is the fine moduli scheme representing the functor of CM-curves with Hilbert polynomial 3t + 1. We show that the moduli scheme of CM-curves in P3 is isomorphic to the twisted cubic component of the Hilbert scheme. We also describe the compactification for twisted cubics in Pn
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| 4. |
- Laksov, Dan, et al.
(författare)
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Notes on flatness and the Quot functor on rings
- 2000
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Ingår i: Communications in Algebra. - : Informa UK Limited. - 0092-7872 .- 1532-4125. ; 28:12, s. 5613-5627
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Tidskriftsartikel (refereegranskat)abstract
- For Rat modules M over a ring A we study the similarities between the three statements, dim(kappa (P))(kappa (P) circle times (A) M) = d far all prime ideals P of A, the A(P)-module Mp is free of rank d for all prime ideals P of A, and M is a locally free A-module of rank d. We have particularly emphasized the case when there is an A-algebra B, essentially of finite type, and M is a finitely generated B-module.
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| 5. |
- Laksov, Dan, et al.
(författare)
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The Hilbert scheme parameterizing finite length subschemes of the line with support at the origin
- 2001
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Ingår i: Compositio Mathematica. - 0010-437X .- 1570-5846. ; 126:3, s. 323-334
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Tidskriftsartikel (refereegranskat)abstract
- We introduce symmetrizing operators of the polynomial ring A[x] in the variable x over a ring A. When A is an algebra over a field k these operators are used to characterize the monic polynomials F(x) of degree n in A[x] such that Ax(k)k[x]((x))/(F(x)) is a free A-module of rank n. We use the characterization to determine the Hilbert scheme parameterizing subschemes of length n of k[x]((x)).
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| 6. |
- Rydh, David, et al.
(författare)
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An intrinsic construction of the principal component of the Hilbert scheme
- 2010
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Ingår i: Journal of the London Mathematical Society. - : Wiley. - 0024-6107 .- 1469-7750. ; 82, s. 459-481
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Tidskriftsartikel (refereegranskat)abstract
- The principal component of the Hilbert scheme of n points on a scheme X is the closure of the open subset parameterizing n distinct points. In this article we construct the principal component as a certain blow-up of the symmetric product of X. Our construction is based on a local explicit analysis of etale families, from which the appropriate universal property, needed to identify the principal component with the blow-up, is derived.
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| 7. |
- Skjelnes, Roy Mikael, et al.
(författare)
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Explicit projective embeddings of standard opens of the Hilbert scheme of points
- 2022
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Ingår i: Journal of Algebra. - : Elsevier BV. - 0021-8693 .- 1090-266X. ; 590, s. 254-276
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Tidskriftsartikel (refereegranskat)abstract
- We describe explicitly how certain standard opens of the Hilbert scheme of points are embedded into Grassmannians. The standard opens of the Hilbert scheme that we consider are given as the intersection of a corresponding basic open affine of the Grassmannian and a closed stratum determined by a Fitting ideal.
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| 8. |
- Skjelnes, Roy Mikael, et al.
(författare)
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Infinite intersections of open subschemes and the Hilbert scheme of points
- 2004
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Ingår i: Journal of Algebra. - : Elsevier BV. - 0021-8693 .- 1090-266X. ; 273:2, s. 571-585
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Tidskriftsartikel (refereegranskat)abstract
- We study infinite intersections of open subschemes and the corresponding infinite intersection of Hilbert schemes. If {U-alpha} is the collection of open subschemes of a variety X containing the fixed point P, then we show that the Hilbert functor of flat and finite families of Spec(O-X,O- P) = boolean AND(alpha) U-alpha is given by the infinite intersection boolean AND(alpha) Hilb(Ualpha), where Hilb(Ualpha), is the Hilbert functor of flat and finite families on U-alpha. In particular, we show that the Hilbert functor of flat and finite families on Spec(O-X,O- P) is representable by a scheme.
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| 9. |
- Skjelnes, Roy Mikael
(författare)
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Nine lines are needed to avoid any three general points
- 2018
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Ingår i: The Mathematics Enthusiast. - : University of Montana - ScholarWorks. - 1551-3440. ; 15:3, s. 407-411
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Tidskriftsartikel (refereegranskat)abstract
- We construct nine lines in the plane over the integers having the property that for any three general points in the plane over any field, at last one of the constructed lines avoids the three points. We then show that nine lines is the fewest number possible having this property.
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| 10. |
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