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3. Other problems

    1. Problem 3.1.

      What is the geography of locally Cohen-Macaulay surfaces in $\mathbb{P}^4$? To be more precise, what numerical invariants (degree, sectional genus, etc.) occur?
        •     The following two problems are about multigraded Hilbert schemes, which parametrize all ideals in a polynomial ring which are homogeneous and have a fixed Hilbert function with respect to a grading by an abelian group [MR2073194] [Haiman and Sturmfels, Multigraded Hilbert schemes. J. Algebraic Geom. 13 (2004), no. 4, 725–769].

          Problem 3.2.

          Is there a multigraded Hilbert scheme with a connected component isomorphic to a fat point?
            • Problem 3.3.

              Is every multigraded Hilbert scheme connected if the polynomial ring is $R=k[x_1,x_2,x_3]$?
                1. Remark. For $R=k[x_1,\dots,x_{26}]$, there are non-connected examples, c.f. [Francisco Santos, Non-connected toric Hilbert-schemes, [MR2181765], [arXiv:0204044]]. Find examples with fewer variables.
                    •     An old question of Joe Harris:

                      Problem 3.4.

                      Does there exist a nondegenerate rigid curve in $\mathbb{P}^n$ other than the rational normal curve?
                          Here rigid means the only deformations of the curve are those induced by automorphisms of $\mathbb{P}^n$.
                        • Problem 3.5.

                          Are there necessary or sufficient conditions on Borel fixed monomial ideals such that they are the generic initial ideals of local Cohen-Macaulay curves?
                            •     Fix a Hilbert polynomial $P$, consider the moduli space $BrV_P{\mathbb{P}^n}$ of branchvarieties that are equidimensional and connected in codimension 1.

                              Problem 3.6.

                              Is $BrV_P(\mathbb{P}^n)$ connected when non-empty?
                                • Problem 3.7.

                                  Is there a rigid local Artinian algebra besides $k^n$?
                                    •     Consider $1, 4, 10, a$ where $6\le a\le 10$. The general Artinian algebra with this Hilbert function is nonsmoothable.

                                      Problem 3.8.

                                      What is the generic point of the component it lies on?

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