3. Other problems

Problem 3.1.
What is the geography of locally CohenMacaulay 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 $R=k[x_1,x_2,x_3]$?
Remark. For $R=k[x_1,\dots,x_{26}]$, there are nonconnected examples, c.f. [Francisco Santos, Nonconnected toric Hilbertschemes, [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? 
Problem 3.5.
Are there necessary or sufficient conditions on Borel fixed monomial ideals such that they are the generic initial ideals of local CohenMacaulay 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 nonempty? 
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.