2
votes
1answer
82 views

Generic vs General property of reducedness in a family of projective schemes

Let $\pi:\mathcal{X} \to B$ be a flat family of projective schemes, $B$ is irreducible. Let $\mathrm{Spec} K$ be a generic point on $B$. Denote by $\mathcal{X}_K$, the pull-back of $\mathcal{X}$. This ...
1
vote
0answers
67 views

Obstruction to Gorenstein Liaisons of space curves

Let $P_1,P$ be Hilbert polynomials of curves in $\mathbb{P}^3$. Denote by $H_{P_1,P}$ the flag Hilbert scheme parametrizing pairs $(C_1 \subset C)$ where $C_1, C$ are of Hilbert polynomials $P_1$ and ...
6
votes
0answers
239 views

A question on infinitesimal deformation (related to intersection theory)

Let $X$ be a connected projective scheme in $\mathbb{P}^n$. Assume, $2 \le \dim X \le n-2$. Let $H$ be a general hyperplane in $\mathbb{P}^n$. Denote by $Z:=X.H$ and $Z'=X.H^m$ for $m \gg 0$. Then ...
1
vote
0answers
85 views

Deformation of complete intersection curves

Let $\mathcal{X} \to B$ be a family of smooth surfaces in $\mathbb{P}^3$. Let $\mathcal{C} \to B$ be a family of curves in $\mathbb{P}^3$. Assume that for all $b \in B$, the fiber $\mathcal{C}_b$ is ...
5
votes
1answer
205 views

A question on the morphism between Hilbert schemes

Let $L_1,L_2$ be two irreducible component of two different Hilbert schemes parametrizing closed subscheme in $\mathbb{P}^n$ and $\mathbb{P}^{n-1}$, respectively. Denote by $\pi_1: \mathcal{X}_1 \to ...
1
vote
1answer
253 views

A question on nested Hilbert scheme

Let $H_1, H_2$ be two Hilbert schemes parametrizing subschemes in $\mathbb{P}^{n_1}, \mathbb{P}^{n_2}$ with Hilbert polynomials $P_1, P_2$, respectively. Given a pair $(Z_1, Z_2)$ of subschemes in ...
3
votes
0answers
253 views

Deformation of a family of curves in a surface

Let $B$ parametrize a family of (reduced) curves in $\mathbb{P}^3$. Assume there exists a smooth hypersurface $X$. in $\mathbb{P}^3$ containing all the curves parametrized by $B$. In otherwords, for ...
1
vote
1answer
102 views

Rational equivalence and infinitesimal deformation of curves

Let $C_1$ and $C_2$ be two rationally equivalent curves in $\mathbb{P}^3$. Is it true that the dimension of $H^0(\mathcal{N}_{C_1|\mathbb{P}^3})$ equal to that of ...
2
votes
0answers
64 views

Changing the Hilbert scheme of curves by adding the hyperplane section

Let $X$ be a smooth degree $d$ ($d>4$) surface in $\mathbb{P}^3$. Let $C$ be a divisor on $X$. Let $D$ be a general element of the linear series $|C+H_X|$ where $H_X$ is the hyperplane section of ...
0
votes
0answers
70 views

Surjectivity on tangent space of Hilbert schemes imply dominance?

Let $P_1, P_2$ be two Hilbert polynomials of certain subschemes in $\mathbb{P}^n$ and $Hilb_{P_1,P_2}$ be the flag Hilbert scheme parametrizing all pairs $(Z_1,Z_2)$ such that $Z_1 \subset Z_2$ and ...
1
vote
0answers
93 views

Functorial property of universal family

Let $P_1, P_2$ and $P_3$ be Hilbert polynomials of projective schemes. Suppose that the flag Hilbert scheme $Hilb_{P_1,P_2}$ is non-empty. Assume further that there exists a closed immersion of the ...
4
votes
1answer
416 views

Is projective morphism with projective fiber flat?

Let $X, Y$ be quasi-projective Noetherian schemes and $f:X \to Y$ be a projective surjective morphism. Assume that every fiber of $f$ is isomorphic to a projective space $\mathbb{P}^n$ for a fixed ...
0
votes
0answers
150 views

Deformation of rational points in a family

Let $\mathcal{X} \to B$ be a family of smooth projective varieties over a field $K$ (possibly finite). Assume that each fiber $\mathcal{X}_b$ of the family has a $K$-rational point. Fix a pair ...
4
votes
1answer
261 views

Complete intersection space curves

Fix two positive integers $d, e$ and assume $d>e$. Is it true that a general degree $e$ curve which lies in a complete intersection of a degree $e$ and a smooth degree $d$ surface in $\mathbb{P}^3$ ...
1
vote
1answer
182 views

Singular locus of a Hilbert scheme

Consider the Hilbert scheme $H$ of conics in $\mathbb{P}^3$. It is easy to see that there exists a closed subscheme $H'$ of $H$ parametrizing $2$ lines intersecting at a point. This can be seen as the ...
1
vote
1answer
97 views

Upper bound on the dimension of linear series on a smooth hypersurface

Let $X$ be a smooth degree $d$ hypersurface in $\mathbb{P}^3$ and $d \ge 5$. Let $C \subset X$ be a reduced curve such that $C$ is not a complete intersection curve in $\mathbb{P}^3$. Is it true that ...
0
votes
0answers
140 views

Normal sheaf of non-reduced locally complete intersection space curves

Let $C$ be a non-reduced locally complete intersection curve on a smooth degree $d$ surface in $\mathbb{P}^3$ (for example a non-reduced Cartier divisor). For simplicity we can assume that ...
1
vote
1answer
452 views

Can any local complete intersection subvariety be an intersection of smooth hypersurfaces

Let $Z$ be a local complete intersection subscheme of dimension $m$ in $\mathbb{P}^{2m+1}$. Let $P$ be the Hilbert polynomial of $Z$. Denote by $Hilb_P$ the Hilbert scheme of local complete ...
3
votes
1answer
242 views

Upper bound on the dimension of the Hilbert scheme of space cuves

Denote by $H_{P,Q}$ the flag Hilbert scheme parametrizing a pair $(C,X)$ such that $X$ is a degree $d$ surface in $\mathbb{P}^3$ with Hilbert polynomial $Q$ and $C \subset X$ is a curve with Hilbert ...
0
votes
1answer
179 views

linear system of non-reduced divisor and associated reduced divisors

Let $X$ be a smooth degree $d$ $(d \ge 5)$ surface in $\mathbb{P}^3$. Let $D$ be an effective Cartier divisor (hence locally of complete intersection) on $X$ and $D_{red}$ the associated reduced ...
3
votes
0answers
237 views

Hilbert function of a Hilbert scheme

Given a Hilbert scheme $H$ of curves in $\mathbb{P}^3$ satisfying certain Hilbert polynomial, is there any way of understanding the degree or arithmetic genus of an irreducible component of the ...
0
votes
1answer
364 views

base-point free linear system

Let $C$ be a reduced curve in a smooth degree $d$ ($d \ge 5$) surface in $\mathbb{P}^3$. Suppose $C=C_1 \cup ... \cup C_r$ with $C_i$ irreducible and $C_i^2<0$ for all $i$. Then is the linear ...
2
votes
0answers
254 views

Hilbert scheme of curves in a degree $d$ surface in $\mathbb{P}^3$

Fix a Hilbert polynomial $P$ of a non-plane curve in $\mathbb{P}^3$. By a curve we mean a reduced scheme of pure dimension $1$ i.e., it can be reducible but is reduced. Suppose that the degree of ...
2
votes
1answer
277 views

Deformation of space curves to union of lines

Does every irreducible component of a Hilbert scheme of curves in $\mathbb{P}^3$ contain a curve that is a union of lines (not necessarily reduced)? Furthermore, given a curve $C \subset \mathbb{P}^3$ ...
0
votes
0answers
138 views

Maximum number of generators of a curve in $\mathbb{P}^3$

Let $H_{d,g}$ denote the Hilbert scheme of curves of degree $d$ and genus $g$ locally of complete intersection in $\mathbb{P}^3$. Given a curve $C \in H_{d,g}$, denote by $S(C)$ a minimal set of ...
1
vote
1answer
178 views

When is the natural projection of the HIlbert flag scheme a flat morphism

Let ${Hilb_{P,Q}}_{red}$ be the reduced scheme associated to the Hilbert flag scheme parametrizing all pairs $(C,X)$ with $C \subset X \subset \mathbb{P}^3$, where $C$ is a curve and $X$ a degree $d$ ...
2
votes
2answers
270 views

General degree $d$ surface in $\mathbb{P}^3$

Let $H_{d_1,g_1}, H_{d_2,g_2}$ be two Hilbert schemes of curves in $\mathbb{P}^3$ with degrees $d_1, d_2$ and genus $g_1, g_2$. Denote by $H:=H_{d_1,g_1}\times H_{d_2,g_2}$ where an element in $H$ is ...
1
vote
2answers
310 views

Any irreducible component of the HIlbert scheme contains an irreducible element

Consider $Hilb_{d,g}$ the Hilbert scheme of curves in $\mathbb{P}^3$ of degree $d$ and genus $g$. Is it true that if $L$ is an irreducible component of $Hilb_{d,g}$ then there exists a curve $C \in L$ ...
5
votes
1answer
625 views

Examples of nice reduced singularities on Hilbert schemes--Edited

In his "Murphy's Law" paper, Vakil showed that every "singularity type" (with a precise meaning) occurs on certain Hilbert schemes; for instance, the Hilbert scheme of nonsingular curves in projective ...
0
votes
1answer
340 views

Irreducible components of the Hilbert scheme

Let $P_1, P_2, Q$ denote the Hilbert scheme of a plane conic in $\mathbb{P}^3$, a quartic and a degree $d$ surface in $\mathbb{P}^3$. Then there is a natural inclusion map $i$ from Hilbert flag scheme ...
1
vote
0answers
154 views

Smooth curve in the Hilbert flag scheme

Let $d$ be an integer greater than $0$. Let $P_2$ be the Hilbert polynomial of a degree $d$ surface in $\mathbb{P}^3$. Recall, the Hilbert flag scheme $\mathrm{Hilb}_{P_1,P_2}$ parametrizes curves $C$ ...
6
votes
2answers
736 views

Relationship between Hilbert schemes and deformation spaces

Hi, I'm just starting to learn about deformation theory (via Hartshorne's Deformation theory, as well as Fantechi's section of FGA explained), and I feel like I'm confused about fundamental concepts. ...