Pathologies on the Hilbert scheme of points

@article{Jelisiejew2019PathologiesOT,
  title={Pathologies on the Hilbert scheme of points},
  author={Joachim Jelisiejew},
  journal={Inventiones mathematicae},
  year={2019},
  volume={220},
  pages={581-610}
}
We prove that the Hilbert scheme of points on a higher dimensional affine space is non-reduced and has components lying entirely in characteristic p for all primes p . In fact, we show that Vakil’s Murphy’s Law holds up to retraction for this scheme. Our main tool is a generalized version of the Białynicki-Birula decomposition. 

Non-reducedness of the Hilbert schemes of few points

We use generalised Białynicki-Birula decomposition, apolarity and obstruction theories to prove non-reducedness of the Hilbert scheme of 13 points on A6. Our argument doesn’t involve computer

Elementary components of Hilbert schemes of points

The Bialynicki-Birula decomposition is generalized to singular schemes and an infinite family of small, elementary and generically smooth components of the Hilbert scheme of points of the affine four-space is found.

Galois closures and elementary components of Hilbert schemes of points

. Bhargava and the first-named author of this paper introduced a functorial Galois closure operation for finite-rank ring extensions, generalizing constructions of Grothendieck and Katz–Mazur. In this

Geometry of canonical genus four curves

We apply Bridgeland stability conditions machinery to describe the geometry of some classical moduli spaces associated with canonical genus four curves in P via an effective control over its

A G ] 1 F eb 2 01 9 Elementary components of Hilbert schemes of points

Consider the Hilbert scheme of points on a higher-dimensional affine space. Its component is elementary if it parameterizes irreducible subschemes. We characterize reduced elementary components in

Rational singularities of nested Hilbert schemes

The Hilbert scheme of pointsHilbpSq of a smooth surface S is a well-studied parameter space, lying at the interface of algebraic geometry, commutative algebra, representation theory, combinatorics,

ON DEFORMATION SPACES OF TORIC VARIETIES

Firstly, we see that the bases of the miniversal deformations of isolated Q-Gorenstein toric singularities are quite restricted. In particular, we classify the analytic germs of embedding dimension ≤

Border Rank of Monomials via Asymptotic Rank

We determine the Waring border rank of monomials. The approach is based on recent results on tensor asymptotic rank which allow us to extend the Ranestad-Schreyer lower bound for the Waring rank of a

On deformation spaces of toric singularities and on singularities of K-moduli of Fano varieties

. Firstly, we see that the bases of the miniversal deformations of isolated Q -Gorenstein toric singularities are quite restricted. In particular, we classify the analytic germs of embedding

Border Waring Rank via Asymptotic Rank

We investigate an extension of a lower bound on the Waring (cactus) rank of homogeneous forms due to Ranestad and Schreyer. We show that for particular classes of homogeneous forms, for which a

References

SHOWING 1-10 OF 36 REFERENCES

Elementary components of Hilbert schemes of points

The Bialynicki-Birula decomposition is generalized to singular schemes and an infinite family of small, elementary and generically smooth components of the Hilbert scheme of points of the affine four-space is found.

Murphy's Law for Hilbert function strata in the Hilbert scheme of points

An open question is whether the Hilbert scheme of points of a high dimensional affine space satisfies Murphy's Law, as formulated by Vakil. In this short note, we instead consider the loci in the

Multigraded Hilbert schemes

We introduce the multigraded Hilbert scheme, which parametrizes all homogeneous ideals with fixed Hilbert function in a polynomial ring that is graded by any abelian group. Our construction is widely

Murphy’s law in algebraic geometry: Badly-behaved deformation spaces

We consider the question: “How bad can the deformation space of an object be?” The answer seems to be: “Unless there is some a priori reason otherwise, the deformation space may be as bad as

On algebraic spaces with an action of G_m

Let Z be an algebraic space of finite type over a field, equipped with an action of the multiplicative group $G_m$. In this situation we define and study a certain algebraic space equipped with an

p-adic derived de Rham cohomology

This paper studies the derived de Rham cohomology of F_p and p-adic schemes, and is inspired by Beilinson's recent work. Generalising work of Illusie, we construct a natural isomorphism between

Le schéma de Hilbert des courbes gauches localement Cohen-Macaulay n'est (presque) jamais réduit

— Let k be an algebraically closed field of characteristic zéro. Let Hd,g dénote thé Hilbert scheme of locally Cohen-Macaulay curves of degree d and genus g contained in thé projective space P|. If d

Representation theory

These are notes from the course MAT4270 on representation theory, autumn 2015. The lectures were held by Sergey Neshyevey. The notes are mine. The first half is about representations of finite

Double-generic initial ideal and Hilbert scheme

Following the approach in the book “Commutative Algebra”, by D. Eisenbud, where the author describes the generic initial ideal by means of a suitable total order on the terms of an exterior power, we

On non-liftable Calabi-Yau threefolds

Only two ways to construct non-liftable Calabi-Yau threefolds are currently known, one example by Hirokado and one method of Schr\"oer. This article computes some cohomological invariants of these