Corpus ID: 119592746

A GIT construction of degenerations of Hilbert schemes of points

@article{Gulbrandsen2016AGC,
  title={A GIT construction of degenerations of Hilbert schemes of points},
  author={Martin G. Gulbrandsen and L. H. Halle and K. Hulek},
  journal={arXiv: Algebraic Geometry},
  year={2016}
}
We present a Geometric Invariant Theory (GIT) construction which allows us to construct good projective degenerations of Hilbert schemes of points for simple degenerations. A comparison with the construction of Li and Wu shows that our GIT stack and the stack they construct are isomorphic, as are the associated coarse moduli schemes. Our construction is sufficiently explicit to obtain good control over the geometry of the singular fibres. We illustrate this by giving a concrete description of… Expand
7 Citations

Figures from this paper

Relative VGIT and an application to degenerations of Hilbert schemes
We generalize the classical semi-continuity theorem for GIT (semi)stable loci under variations of linearizations to a relative situation of an equivariant projective morphism from X to an affine baseExpand
Symmetric products of a semistable degeneration of surfaces
We explicitly construct a V-normal crossing Gorenstein canonical model of the relative symmetric products of a local semistable degeneration of surfaces without a triple point by means of toricExpand
Gulbrandsen-Halle-Hulek degeneration and Hilbert-Chow morphism
For a semistable degeneration of surfaces without a triple point, we show that two models of degeneration of Hilbert scheme of points of the family, Gulbrandsen-Halle-Hulek degeneration given inExpand
Remarks on degenerations of hyper-K\"ahler manifolds
Using the Minimal Model Program, any degeneration of K-trivial varieties can be arranged to be in a Kulikov type form, i.e. with trivial relative canonical divisor and mild singularities. In theExpand
The geometry of degenerations of Hilbert schemes of points
Given a strict simple degeneration $f \colon X\to C$ the first three authors previously constructed a degeneration $I^n_{X/C} \to C$ of the relative degree $n$ Hilbert scheme of $0$-dimensionalExpand
An inclusion-exclusion principle for tautological sheaves on Hilbert scheme of points
We prove an equation of Euler characteristics of tautological sheaves on the Hilbert scheme of points on the fibers of a simple degeneration.
The essential skeleton of a product of degenerations
We study the problem of how the dual complex of the special fiber of a strict normal crossings degeneration $\mathscr{X}_{R}$ changes under products. We view the dual complex as a skeleton inside theExpand

References

SHOWING 1-10 OF 18 REFERENCES
On monodromies of a degeneration of irreducible symplectic Kähler manifolds
We study the monodromy operators on the Betti cohomologies associated to a good degeneration of irreducible symplectic manifold and we show that the unipotency of the monodromy operator on the middleExpand
The geometry of moduli spaces of sheaves
Preface to the second edition Preface to the first edition Introduction Part I. General Theory: 1. Preliminaries 2. Families of sheaves 3. The Grauert-Mullich Theorem 4. Moduli spaces Part II.Expand
Good degeneration of Quot-schemes and coherent systems
We construct good degenerations of Quot-schemes and coherent systems using the stack of expanded degenerations. We show that these good degenerations are separated and proper DM stacks of finiteExpand
A relative Hilbert–Mumford criterion
We generalize the classical Hilbert–Mumford criteria for GIT (semi-)stability in terms of one parameter subgroups of a linearly reductive group G over a field k, to the relative situation of anExpand
Geometric Invariant Theory
“Geometric Invariant Theory” by Mumford/Fogarty (the first edition was published in 1965, a second, enlarged edition appeared in 1982) is the standard reference on applications of invariant theory toExpand
Equivariant versal deformations of semistable curves
We prove that given any $n$-pointed prestable curve $C$ of genus $g$ with linearly reductive automorphism group ${\rm Aut}(C)$, there exists an ${\rm Aut}(C)$-equivariant miniversal deformation ofExpand
Configurations of points on degenerate varieties and properness of moduli spaces
Consider a smooth variety $X$ and a smooth divisor $D\subset X$. Kim and Sato (arXiv:0806.3819) define a natural compactification of $(X\setminus D)^n$, denoted $X_D^{[n]}$, which is a moduli spaceExpand
Stable Morphisms to Singular Schemes and Relative Stable Morphisms
Algebraic Geometry
Introduction to Algebraic Geometry.By Serge Lang. Pp. xi + 260. (Addison–Wesley: Reading, Massachusetts, 1972.)
Good degenerations of moduli spaces
  • In Handbook of moduli. Vol. II, volume 25 of Adv. Lect. Math. (ALM),
  • 2013
...
1
2
...