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  • 1.
    Ekedahl, Torsten
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Skjelnes, Roy
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Recovering the good component of the Hilbert scheme2014In: Annals of Mathematics, ISSN 0003-486X, E-ISSN 1939-8980, Vol. 179, no 3, p. 805-841Article in journal (Refereed)
    Abstract [en]

    We give an explicit construction, for a flat map X -> S of algebraic spaces, of an ideal in the n'th symmetric product of X over S. Blowing up this ideal is then shown to be isomorphic to the schematic closure in the Hilbert scheme of length n subschemesof the locusof n distinct points.This generalizes Haiman's corresponding result for the affine complex plane.However, our construction of the ideal is very different from that of Haiman, using the formalism of divided powers rather than representation theory.In the nonflat case we obtain a similar result by replacing the n'th symmetric product by the n'th divided power product.

  • 2. Gustavsen, T. S.
    et al.
    Laksov, Dan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Skjelnes, Roy Mikael
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    An elementary, explicit, proof of the existence of Hilbert schemes of points2007In: Journal of Pure and Applied Algebra, ISSN 0022-4049, E-ISSN 1873-1376, Vol. 210, no 3, p. 705-720Article in journal (Refereed)
    Abstract [en]

    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.

  • 3. Gustavsen, Trond Stolen
    et al.
    Laksov, Dan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Skjelnes, Roy Mikael
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    An elementary, explicit, proof of the existence of quot schemes of points2007In: Pacific Journal of Mathematics, ISSN 0030-8730, E-ISSN 1945-5844, Vol. 231, no 2, p. 401-415Article in journal (Refereed)
    Abstract [en]

    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.

  • 4.
    Laksov, Dan
    et al.
    KTH, Superseded Departments, Mathematics.
    Pitteloud, Y.
    Skjelnes, Roy Mikael
    KTH, Superseded Departments, Mathematics.
    Notes on flatness and the Quot functor on rings2000In: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 28, no 12, p. 5613-5627Article in journal (Refereed)
    Abstract [en]

    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.

  • 5.
    Laksov, Dan
    et al.
    KTH, Superseded Departments, Mathematics.
    Skjelnes, Roy Mikael
    KTH, Superseded Departments, Mathematics.
    The Hilbert scheme parameterizing finite length subschemes of the line with support at the origin2001In: Compositio Mathematica, ISSN 0010-437X, E-ISSN 1570-5846, Vol. 126, no 3, p. 323-334Article in journal (Refereed)
    Abstract [en]

    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)).

  • 6.
    Lundkvist, Christian
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Skjelnes, Roy
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Non-effective deformations of Grothendieck’s Hilbertfunctor2008In: Mathematische Zeitschrift, ISSN 0025-5874, E-ISSN 1432-1823, Vol. 258, no 3, p. 513-519Article in journal (Refereed)
    Abstract [en]

    Let X be a scheme that does not satisfy the valuativecriterion of separatedness. We show that the Hilbert functorparametrizing closed families of X that are flat, finite and of rankone is not represented by a scheme or an algebraic space.

  • 7.
    Rydh, David
    et al.
    Department of Mathematics, University of California.
    Skjelnes, Roy Mikael
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    An intrinsic construction of the principal component of the Hilbert scheme2010In: Journal of the London Mathematical Society, ISSN 0024-6107, E-ISSN 1469-7750, Vol. 82, p. 459-481Article in journal (Refereed)
    Abstract [en]

    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.

  • 8.
    Skjelnes, Roy
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Computing Hasse-Schmidt derivations and Weil restrictions over jets2014In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 411, p. 114-128Article in journal (Refereed)
    Abstract [en]

    We give an explicit and compact description of the Hasse-Schmidt derivations, using Fitting ideals and symmetric tensor algebras. Finally we verify the localization conjecture.

  • 9.
    Skjelnes, Roy
    KTH, Superseded Departments, Mathematics.
    Symmetric tensors with applications to Hilbert schemes1999Article in journal (Other academic)
    Abstract [en]

    Let A[X]_U be a fraction ring of the polynomial ring A[X] in the variable X over a commutative ring A. We show that the Hilbert functor {Hilb}^n_{A[X]_U} is represented by an affine scheme SymmnA(A[X]U) give as the ring of symmetric tensors of ⊗nAA[X]U . The universal family is given as Symmn−1A(A[X]U)×ASpec(A[X]U) .

  • 10.
    Skjelnes, Roy M.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Weil restriction and the Quot scheme2015In: Algebraic Geometry, ISSN 2313-1691, E-ISSN 2214-2584, Vol. 2, no 4, p. 514-534Article in journal (Refereed)
    Abstract [en]

    We introduce a concept that we call module restriction, which generalizes the classical Weil restriction. After having established some fundamental properties as existence and étaleness, we apply our results to show that the Quot functor Quotn FX/S of Grothendieck is representable by an algebraic space for any quasi-coherent sheaf FX on any separated algebraic space X → S.

  • 11.
    Skjelnes, Roy Mikael
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Nine lines are needed to avoid any three general points2018In: The Montana Mathematics Enthusiast, ISSN 1551-3440, E-ISSN 1551-3440, Vol. 15, no 3, p. 407-411Article in journal (Refereed)
    Abstract [en]

    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.

  • 12.
    Skjelnes, Roy Mikael
    KTH, Superseded Departments, Mathematics.
    On Hilbert schemes parameterizing points on the affine line having support in a fixed subset2000Doctoral thesis, comprehensive summary (Other scientific)
  • 13.
    Skjelnes, Roy Mikael
    KTH, Superseded Departments, Mathematics.
    On the representability of H ilb(n)k x ((x))2000In: Journal of the London Mathematical Society, ISSN 0024-6107, E-ISSN 1469-7750, Vol. 62, p. 757-770Article in journal (Refereed)
    Abstract [en]

    Let k[x]((x)) be the polynomial ring k[x] localized in the maximal ideal (x) subset of k[x]. The paper studies the Hilbert functor parametrizing ideals of colength n in this ring having support at the origin. The main result is that this functor is not representable. A complete description of the functor as a limit of representable functors is also given.

  • 14.
    Skjelnes, Roy Mikael
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Quot Schemes in Grassmannians2018In: Indiana University Mathematics Journal, ISSN 0022-2518, E-ISSN 1943-5258, Vol. 67, no 2, p. 623-643Article in journal (Refereed)
    Abstract [en]

    We study the Quot scheme parametrizing constant Hilbert polynomial quotients of the free sheaf of rank p on projective space. Our main result describes the closed strata that defines these Quot schemes as closed subschemes in Grassmannians.

  • 15.
    Skjelnes, Roy Mikael
    KTH, Superseded Departments, Mathematics.
    Resultants and the Hilbert scheme of points on the line2002In: Arkiv för matematik, ISSN 0004-2080, E-ISSN 1871-2487, Vol. 40, no 1, p. 189-200Article in journal (Refereed)
    Abstract [en]

    We present an elementary and concrete description of the Hilbert scheme of points on the spectrum of fraction rings k[X](U) of the one-variable polynomial ring over a commutative ring k. Our description is based on the computation of the resultant of polynomials in k[X]. The present paper generalizes the results of Laksov-Skjelnes [7], where the Hilbert scheme, on spectrum of the local ring of a point was described.

  • 16.
    Skjelnes, Roy Mikael
    et al.
    KTH, Superseded Departments, Mathematics.
    Walter, C.
    Infinite intersections of open subschemes and the Hilbert scheme of points2004In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 273, no 2, p. 571-585Article in journal (Refereed)
    Abstract [en]

    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.

1 - 16 of 16
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