All Questions
18 questions with no upvoted or accepted answers
6
votes
0
answers
320
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 ...
- 833
4
votes
0
answers
362
views
Is complete intersection a open or closed property in Hilbert schemes
Fix an integer $N$, $X$ a (smooth) complete intersection subvariety in $\mathbb{P}^N$. Denote by $P$ the Hilbert polynomial of $X$ (as a subvariety in $\mathbb{P}^N$). Consider the Hilbert scheme $\...
- 2,126
3
votes
0
answers
222
views
Deformations of genus g curves to 'non-reduced rational curve'
We work over the complex numbers. Fix a genus $g$. Does there exist a connected reduced base $ B $ and a flat projective family $ \pi : X \rightarrow B $ satisfying the following two conditions?
its ...
- 936
3
votes
0
answers
456
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 ...
- 1,040
3
votes
0
answers
312
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 ...
- 1,040
3
votes
0
answers
250
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 $C$ ...
- 1,040
2
votes
0
answers
124
views
Consequences of smoothability
I have seen that there is a lot of work on studying the smoothable component of the Hilbert scheme of points $\textit{Hilb}^n(X)$ of some variety $X$. The main results are that if $\dim X \leq 2$ then ...
2
votes
0
answers
153
views
An analogue of Brill-Noether for hypersurfaces?
Let $d,g,r$ be natural numbers such that $d \geq 1$, $g \geq 2$, $r \geq 3$. Denote by $\mathcal{H}_{d,g,r}$ the Hilbert scheme classifying subschemes of $\mathbb{P}^r$ with Hilbert polynomial $P(x) = ...
- 1,981
2
votes
0
answers
357
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 ...
- 833
2
votes
0
answers
108
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 $...
- 1,040
2
votes
0
answers
451
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 ...
- 1,040
1
vote
0
answers
239
views
Proposition from Kollar's Rational Curves on Algebraic Varieties
$\DeclareMathOperator\Hom{Hom}$I have some questions on a proof of Proposition II.3.10 & notations from Rational Curves
on Algebraic Varieties by Janos Kollar (page 117).
We work in setting ...
- 6,028
1
vote
0
answers
310
views
Hilbert scheme of Grassmannians
Let $X=\mathbb{G}(k,n)$ be the Grassmannian of $k$-planes in $n$-space. Let $Y\subseteq X$ a subvariety and let $H$ be the connected component of the Hilbert scheme of $X$ that contains $Y$.
Is $H$ ...
- 3,031
1
vote
0
answers
125
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 $...
- 503
1
vote
0
answers
132
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 ...
- 1,593
0
votes
0
answers
214
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 $(p,\...
- 2,022
0
votes
0
answers
401
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 $d>deg(C)...
- 1,040
0
votes
0
answers
164
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 ...
- 1,040