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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.
    1.     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 ${\rm Gal}(\overline{\mathbb{F}}_p/\mathbb{F}_p)$.
        1. 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 Cohen-Macaulay 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?
                    1. 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 $e$.
                            1. 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 ${\rm Hilb}_{d,g}(\mathbb{P}^n)$.
                                      For $\mathbb{P}^3$ a lower bound is $4d$; for $\mathbb{P}^4$ the lower bound is $5d+1-g$ which is obviously not a good bound for $d$ fixed and $g$ sufficiently large.
                                    1. 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 Cohen-Macaulay?
                                            1. 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?
                                                    1. Remark. This is a hard problem. If the curves are arithmetically Cohen-Macaulay, 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.
                                                            1. 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]].
                                                                • Hartshorne-Rao modules

                                                                      The following three problems are related to the Hartshorne-Rao 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 Hartshorne-Rao 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. Martin-Deschamps, D. Perrin, Sur la classification des courbes gauches. Astérisque No. 184-185 (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?
                                                                    1. Remark. It is proved that if $4q<\max(6p+r,6r+p)$ then $E_{p,q,r}$ is reducible ([MR1219697] [M. Martin-Deschamps, 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 2.5.

                                                                          Describe the irreducible component of $E_\rho$.
                                                                            • 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?
                                                                                1. Remark. We might have to require $C$ to be minimal.

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