1. Hilbert schemes of points
Denote by ${\rm Hilb}^d(\mathbb{A}^n)$ the Hilbert scheme of $d$ points in the affine space $\mathbb{A}^n$. By component we always mean an irreducible component.
Problem 1.05.
Describe the singularities of the smoothable component of ${\rm Hilb}^d(\mathbb{A}^n)$. To be more specific:
How large can the dimension of the Zariski tangent space to this component get?
Does the maximum occur in the intersection of components? Does it occur in the smoothable component? 
Problem 1.1.
Can you describe the Zariski tangent space to the smoothable component of ${\rm Hilb}^d(\mathbb{A}^n)$? 
Problem 1.15.
What is the smallest $d$ such that ${\rm Hilb}^d(\mathbb{A}^3)$ is reducible?
Remark. We know that $10 < d \le 78$. The bound $d>10$ needs the assumption that char=0 and follows from a recent paper of Sivic.


Problem 1.2.
(1) Is there a component of ${\rm Hilb}^d(\mathbb{A}^n)$ that exists only in characteristic $p$ for some $p$?
(2) Same question for the Hilbert schemes of curves in $\mathbb{P}^3$. 
Problem 1.25.
Is there a nonreduced component of ${\rm Hilb}^d(\mathbb{A}^n)$? If so, find it. 
Problem 1.3.
(1) Give an explicit example (or show it does not happen) of a geometrically irreducible component of ${\rm Hilb}^d(\mathbb{A}^n)$, which is not fixed under the action of ${\rm Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$.
(2) Same question for the Hilbert schemes of curves in $\mathbb{P}^3$. 
Problem 1.35.
Is the Gröbner fan a discrete invariant that distinguishes the components of ${\rm Hilb}^d(\mathbb{A}^n)$? 
Problem 1.45.
If a component of the Hilbert scheme contains a smooth Borel fixed point, does the component have to be rational?
Remark. If the ideal is a segment ideal ( i.e. a monomial ideal generated in some degree $s$, where $s$ is not larger than its regularity, by the maximal monomials with respect to some term ordering), then the component is known to be rational, c.f. [P. Lella, M. Roggero, Rational components of Hilbert schemes, [arXiv:0903.1029]]. Note that every segment ideal is a Borel fixed ideal, but the converse is not true.


Let $C$ be a curve.
Problem 1.5.
Can ${\rm Hilb}^d(C)$ have a component of dimension less than $d$?
Remark. There exists a nonsmoothable component of $dim=d>1$.


Problem 1.55.
Give a geometric algebraic description of generic points of irreducible components of ${\rm Hilb}^d(\mathbb{A}^n)$. 
Problem 1.6.
Is ${\rm Hilb}^8(\mathbb{A}^4)$ reduced? More generally, develop techniques to prove the reducedness of Hilbert schemes. 
Problem 1.65.
Is it possible to define analogues of the Nakajima operators for the cohomology of a desingularization of the smoothable component of ${\rm Hilb}^d(\mathbb{A}^n)$? 
Problem 1.7.
Does the set of monomial ideals contained in a component of ${\rm Hilb}^d(\mathbb{A}^n)$ determine that component? (Or more generally for ${\rm Hilb}^P(\mathbb{P}^n)$ where the Hilbert polynomial $P$ is arbitrary.)
Remark. For the analogous question for the Hilbert scheme of curves, Richard Liebling might have given a counterexample in his thesis.

Cite this as: AimPL: Components of Hilbert schemes, available at http://aimpl.org/hilbertschemes.