2. Hilbert schemes of curves
Denote by ${\rm Hilb}_{d,g}(\mathbb{P}^r)$ the Hilbert scheme of curves of degree $d$ and genus $g$ in the projective space $\mathbb{P}^r$. By "component" we always mean an irreducible component.
Fix $d, g, r, e>0$. Let ${\rm Hilb}^{sm}_{d,g}(\mathbb{P}^r)$ be the open subscheme of the Hilbert scheme ${\rm Hilb}_{d,g}(\mathbb{P}^r)$ that parameterizes smooth curves. For each point $[C]\in {\rm Hilb}^{sm}_{d,g}(\mathbb{P}^r)$, we define the Gauss map $C \to {\rm Gr}(1,r)$ sending a point of $C$ to the tangent line at that point. Define $$Z^e:=\{[C]\in{\rm Hilb}^{sm}_{d,g}(\mathbb{P}^r) \textrm{ the Gauss map of the curve $C$ is inseparable of degree $p^e$}\}.$$ Then $\bigcup_{e\ge 0}Z^e={\rm Hilb}^{sm}_{d,g}(\mathbb{P}^r)$.
Problem 2.05.
What can we say about the set $Z^e$? Can we construct exotic components (i.e. components that only exist in characteristic $p$) using this stratification? Study the action by the Galois group ${\rm Gal}(\overline{\mathbb{F}}_p/\mathbb{F}_p)$.
Remark. There are some other stratifications one may consider, e.g. the one obtained by the topological type of the ramification divisor.


Problem 2.1.
Is the Hilbert scheme of local CohenMacaulay curves in $\mathbb{P}^3$ connected? 
Let $\mathcal{C}_t$ be a family of curves in $\mathbb{P}^3$ such that a general curve in this family is a smooth complete intersection, and the special curve $C_0$ is smooth.
Problem 2.15.
Is $C_0$ always a complete intersection, assuming that the characteristic is 0?
Remark. If the characteristic is $p>0$, the answer is negative; if $n>3$, the answer is negative; if $C_0$ is not smooth, the answer is negative. The reference is [MR1702099][P. Ellia, R. Hartshorne, Smooth specializations of space curves: questions and examples, Commutative algebra and algebraic geometry (Ferrara), 53–79, Lecture Notes in Pure and Appl. Math., 206, Dekker, New York, 1999].


Let $S\subset \mathbb{P}^3$ be an integral surface of degree $d$ with a double curve $D$ of degree $e$, and triple points, pinch point, etc.
Conjecture 2.2.
[P. Ellia and R. Hartshorne] There exists $n_0\in\mathbb{Z}$ such that for any smooth curve $C\subset S$ of degree $n\ge n_0$, let $H_C$ be the irreducible component of the Hilbert scheme containing $C$, then a general $C'\in H_C$ is also contained in a surface $S'\subset \mathbb{P}^3$ of degree $d$ and a double curve $D'$ of degree $e$.
Remark. The conjecture is posed in [MR1702099][P. Ellia, R. Hartshorne, Smooth specializations of space curves: questions and examples, Commutative algebra and algebraic geometry (Ferrara), 53–79, Lecture Notes in Pure and Appl. Math., 206, Dekker, New York, 1999]. It is proved to be true for $d=3$ in [MR2010789][J. Brevik, F. Mordasini, Curves on a ruled cubic surface, Collect. Math. 54 (2003), no. 3, 269–281].


Problem 2.25.
Fix $d,g,n$. Find a good/sharp lower bound for the dimension of components of ${\rm Hilb}_{d,g}(\mathbb{P}^n)$.
Remark. In the range $d^3\le \lambda(n) g^2$ there is a better bound, c.f. [MR2576683][D. Chen, On the dimension of the Hilbert scheme of curves, Math. Res. Lett. 16 (2009), no. 6, 941–954. Theorem 1.3].


Problem 2.3.
(1) Let $C$ be of bidegree $(3,7)$ on a nonsingular quadric surface in $\mathbb{P}^3$. Can $C$ be connected to an extremal curve in ${\rm Hilb}_{10,12}(\mathbb{P}^3)$?
(2) Given $4$ skew lines $C_1$ on a nonsingular quadric $Q_1$ in $\mathbb{P}^3$. Does there exists a family $Q_t\rightsquigarrow 2H$, and a family $\mathcal{C}_t\subset Q_t$ such that $C_0\subset Q_0=2H$ is locally CohenMacaulay?
Remark. (2) implies (1).


Fix $d,g$. Let $H\subset{\rm Hilb}_{d,g}(\mathbb{P}^3)$ be an irreducible component.
Problem 2.35.
What is the largest number of points in general position you can make these curves pass through?
Remark. This is a hard problem. If the curves are arithmetically CohenMacaulay, it should be treatable.


Problem 2.4.
What is the best pair $(d,g)$ (in the vague sense that $d$ is as small as possible and $g$ is close to $0$ as possible) such that there is a component of ${\rm Hilb}_{d,g}(\mathbb{P}^3)$ whose general member has an embedded point.
Remark. An example of a pair satisfying the above condition is $d=4$, $g=15$, c.f. [D. Chen, S. Nollet, Detaching embedded points, [arXiv:0911.2221]].


HartshorneRao modules
The following three problems are related to the HartshorneRao modules. Let $\mathbb{P}^3$ be a projective space with homogeneous coordinate ring $R=k[X,Y,Z,T]$. Given a space curve $C$ with ideal sheaf $\mathcal{I}_C$, the HartshorneRao module is defined as the $R$module $M_C=\oplus_{i\in\mathbb{Z}}H^1\mathcal{I}_C(i)$. A program to study the Hilbert scheme $H_{d,g}$ of space curves of degree $d$ and genus $g$ is outlined in [MR1073438][M. MartinDeschamps, D. Perrin, Sur la classification des courbes gauches. Astérisque No. 184185 (1990), 208 pp.]. Namely, $H_{d,g}$ can be stratified into $H_{\gamma,\rho}$ consisting of curves $C$ such that, for every $i\in\mathbb{Z}$, $H^0\mathcal{I}_C(i)$ and $H^1\mathcal{I}_C(i)$ have fixed dimensions $\gamma(i)$ and $\rho(i)$, respectively. Now let $E_\rho$ be the moduli space of finite length graded modules with Hilbert function $\rho$. There is a natural map from $H_{\gamma,\rho}$ to $E_\rho$, and properties of $H_{d,g}$ can be extracted from properties of $E_\rho$.Problem 2.45.
Let $\rho=(p,q,r,0,0,\dots)$ and define $E_{p,q,r}:=E_\rho$. For which $p, q, r$ is $E_{p,q,r}$ irreducible?
Remark. It is proved that if $4q<\max(6p+r,6r+p)$ then $E_{p,q,r}$ is reducible ([MR1219697] [M. MartinDeschamps, D. Perrin, Courbes gauches et modules de Rao, J. Reine Angew. Math. 439 (1993), 103–145. page 119, Theorem 2.1]). Conjecturally, if $4q\ge \max(6p+r,6r+p)$ then $E_{p,q,r}$ is irreducible.


Problem 2.55.
What do properties of the Rao modules imply about $C$? For example, if $M_C$ is Gorenstein or annihilated by a linear form, does $C$ have any nice properties?
Remark. We might have to require $C$ to be minimal.

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