| Register
\(\newcommand{\Cat}{{\rm Cat}} \) \(\newcommand{\A}{\mathcal A} \) \(\newcommand{\freestar}{ \framebox[7pt]{$\star$} }\)

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.
    1. 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?
                1. 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 Gro╠łbner fan a discrete invariant that distinguishes the components of ${\rm Hilb}^d(\mathbb{A}^n)$?
                                    • Problem 1.4.

                                      Does there exist a nonrational component 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?
                                            1. 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 $d$?
                                                    1. Remark. There exists a non-smoothable 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.)
                                                                        1. 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.