arXiv subjects

Changho Keem

Publications and source records attributed to Changho Keem.

14 recordsLinked to original sources

arXiv2026

Hilbert scheme of smooth projective curves of unexpected dimension \& existence of a component with less than the expected number of moduli

We denote by $\mathcal{H}_{d,g,r}$ the Hilbert scheme of smooth curves of degree $d$ and genus $g$ in $\mathbb{P}^r$. Denoting by $\mathcal{M}_g$ the moduli space of smooth curves of genus $g$, let $\mu: \mathcal{H}_{d,g,r}\dasharrow \mathcal{M}_g$ be the natural map sending $X\in\mathcal{H}_{d,g,r}$ to its isomorphism class $\mu (X)=[X]\in\mathcal{M}_g$. It has been conjectured that a component $\mathcal{H}\subset\mathcal{H}_{d,g,r}$ has the minimal possible dimension $$\Xx(d,g,r):=3g-3+\rho(d,g,r)+\dim\operatorname{Aut}(\mathbb{P}^r)$$ \noindent if $\codim_{\mathcal{M}_g}\mu(\mathcal{H})\le g-5$ provided $\Xx(d,g,r)\ge 0$, where $\rho(d,g,r):=g-(r+1)(g-d+r)$ is the Brill-Noether number. In this article, we exhibit examples against the conjecture discuss further for the study of the functorial map $\mu: \\mathcal{H}{d,g,r}\dasharrow\mathcal{M}_g$ along this line. A component $\mathcal{H}\subset \mathcal{H}_{d,g,r}$ is said to have the {\it expected number of moduli} if $$\dim\mu({\Hh})=\min\{3g-3, 3g-3+\rho(d,g,r)\},$$provided $3g-3+\rho(d,g,r)\ge 0$. The existence of a component with strictly less than the expected number of moduli has not been known. In this paper, we show the existence of components with less than the expected number of moduli.

math.AG↗

arXiv2025

Hilbert scheme and Hilbert functions of smooth curves of degrees at most $15$ in $\mathbb{P}^5$

Denoting $\mathcal{H}_{d,g,5}$ by the Hilbert scheme of smooth curves of degree $d$ and genus $g$ in $\mathbb{P}^5$, let $\mathcal{H}$ be an irreducible component of $\mathcal{H}_{d,g,5}$. We study the Hilbert function $h_X:\mathbb{N}\longrightarrow\mathbb{N}$, $h_X(t):= h^0(\mathcal{I}_X(t))$ of a general member $X\in\mathcal{H}$, especially when the degree of the curve is low; $d\le 15$. We also determine the irreducibility of $\mathcal{H}_{d,g,5}$ for $d\le 14$ and study the natural functorial map $\mu :$\mathcal{H}_{d,g,5}$ \longrightarrow \mathcal{M}_g$ in some detail. We describe the fibre $\mu^{-1}\mu(X)$ for a general $X\in\mathcal{H} $ as well as determining the projective normality (or being ACM).

math.AG↗

arXiv2025

Hilbert scheme of smooth curves of degree sixteen in $\mathbb{P}^5$

We denote by $\mathcal{H}_{d,g,r}$ the Hilbert scheme of smooth curves, which is the union of components whose general point corresponds to a smooth irreducible and non-degenerate curve of degree $d$ and genus $g$ in $\mathbb{P}^r$. In this article, we study $\mathcal{H}_{16,g,5}$ for almost every possible genus $g$ and chasing after its irreducibility. We also study the natural moduli map $\mathcal{H}_{d,g,5}\stackrel{\mu}{\to}\mathcal{M}_g$ and several key properties such as gonality of a general element as well as characterizing smooth elements in each component.

math.AG↗

arXiv2024

Hilbert scheme of linearly normal curves in $\mathbb{P}^r$ with index of speciality five and beyond

We study the Hilbert scheme of smooth, irreducible, non-degenerate and linearly normal curves of degree $d$ and genus $g$ in $\mathbb{P}^r$ ($r\ge 3$) whose complete and very ample hyperplane linear series $\mathcal{D}$ have relatively small index of speciality $i(\mathcal{D})=g-d+r$. In particular we determine the existence as well as the non-existence of Hilbert schemes of linearly normal curves $\mathcal{H}^{\mathcal{L}}_{d,g,r}$ for every possible triples $(d,g,r)$ with $i(\mathcal{D})=5$ and $r\ge 3$. We also determine the irreducibility of the Hilbert scheme $\mathcal{H}^{\mathcal{L}}_{g+r-5,g,r}$ when the genus $g$ is near to the minimal possible value with respect to the dimension of the projective space $\mathbb{P}^r$ for which $\mathcal{H}^{\mathcal{L}}_{g+r-5,g,r}\neq\emptyset$, say $r+9\le g\le r+11$. In the course of proofs of key results, we show the existence of linearly normal curves of degree $d\ge g+1$ with arbitrarily given index of speciality with some mild restriction on the genus $g$.

math.AG↗

arXiv2023

On the Hilbert scheme of smooth curves of degree $15$ and genus $14$ in $\mathbb{P}^5$

We denote by $\mathcal{H}_{d,g,r}$ the Hilbert scheme of smooth curves, which is the union of components whose general point corresponds to a smooth irreducible and non-degenerate curve of degree $d$ and genus $g$ in $\mathbb{P}^r$. In this article, we show that $\mathcal{H}_{15,14,5}$ is non empty and reducible with two components of the expected dimension hence generically reduced. We also study the birationality of the moduli map up to projective motion and several key properties such as gonality of a general element as well as specifying smooth elements of each components.

math.AG↗

arXiv2023

On the Hilbert scheme of smooth curves of degree $d=15$ in $\mathbb{P}^5$

We denote by $\mathcal{H}_{d,g,r}$ the Hilbert scheme of smooth curves, which is the union of components whose general point corresponds to a smooth, irreducible and non-degenerate curve of degree $d$ and genus $g$ in $\mathbb{P}^r.$ In this article, we study $\mathcal{H}_{15,g,5}$ for every possible genus $g$ and determine when it is irreducible. We also study the moduli map $\mathcal{H}_{15,g,5}\rightarrow\mathcal{M}_g$ and several key properties such as gonality of a general element as well as characterizing smooth elements of each component.

math.AG↗

arXiv2022

Existence and the reducibility of the Hilbert scheme of linearly normal curves in $\mathbb{P}^r$ of relatively high degrees

Let $\mathcal{H}_{d,g,r}$ be the Hilbert scheme parametrizing smooth irreducible and non-degenerate curves of degree $d$ and genus $g$ in $\mathbb{P}^r.$ We denote by $\mathcal{H}^\mathcal{L}_{d,g,r}$ the union of those components of $\mathcal{H}_{d,g,r}$ whose general element is linearly normal. In this article we show that $\mathcal{H}^\mathcal{L}_{d,g,r}$ ($d\ge g+r-3$) is non-empty in a certain optimal range of triples $(d,g,r)$ and is empty outside the range. This settles the existence (or non-emptiness if one prefers) of the Hilbert scheme $\mathcal{H}^\mathcal{L}_{d,g,r}$ of linearly normal curves of degree $d$ and genus $g$ in $\mathbb{P}^r$ for $g+r-3\le d\le g+r$, $r\ge 3$. We also determine all the triples $(d,g,r)$ with $g+r-3\le d\le g+r$ for which $\mathcal{H}^\mathcal{L}_{d,g,r}$ is reducible (or irreducible).

math.AG↗

arXiv2022

On the Hilbert scheme of linearly normal curves in $\mathbb{P}^r$ with small index of speciality

We study the Hilbert scheme $\mathcal{H}^\mathcal{L}_{d,g,r}$ parametrizing smooth, irreducible, non-degenerate and linearly normal curves of degree $d$ and genus $g$ in $\mathbb{P}^r$ whose complete and very ample hyperplane linear series $\mathcal{D}$ have relatively small index of speciality $i(\mathcal{D})=g-d+r$. In particular we show the existence (and non-existence as well in some sporadic cases) of every Hilbert scheme of linearly normal curves with $i(\mathcal{D})=4$. We also determine the irreducibility of $\mathcal{H}^\mathcal{L}_{2r+4,r+8,r}$ for $3\le r\le 8$, which are rather peculiar families in a certain sense.

math.AG↗

arXiv2020

On the Hilbert scheme of smooth curves in $\mathbb{P}^4$ of degree $d = g+1$ and genus $g$ with negative Brill-Noether number

We denote by $\mathcal{H}_{d,g,r}$ the Hilbert scheme of smooth curves, which is the union of components whose general point corresponds to a smooth irreducible and non-degenerate curve of degree $d$ and genus $g$ in $\PP^r$. In this article, we show that for low genus $g$ outside the Brill-Noether range, the Hilbert scheme $\mathcal{H}_{g+1,g,4}$ is non-empty whenever $g\ge 9$ and irreducible whose only component generically consists of linearly normal curves unless $g=9$ or $g=12$. This complements the validity of the original assertion of Severi regarding the irreducibility of $\mathcal{H}_{d,g,r}$ outside the Brill-Nother range for $d=g+1$ and $r=4$.

math.AG↗

arXiv2019

On the Hilbert scheme of linearly normal curves in $\mathbb{P}^r$ of relatively high degree

Let $\mathcal{H}_{d,g,r}$ be the Hilbert scheme parametrizing smooth irreducible and non-degenerate curves of degree $d$ and genus $g$ in $\PP^r$. We denote by $\mathcal{H}^\mathcal{L}_{d,g,r}$ the union of those components of $\mathcal{H}_{d,g,r}$ whose general element is linearly normal and we show that any non-empty $\mathcal{H}^\mathcal{L}_{d,g,r}$ ($d\ge g+r-3$) is irreducible for an extensive range of triples $(d,g,r)$ beyond the Brill-Noether range. This establishes the validity of a suitably modified assertion of Severi regarding the irreducibility of the Hilbert scheme $\mathcal{H}^\mathcal{L}_{d,g,r}$ of linearly normal curves for $g+r-3\le d\le g+r$, $r\ge 3$, and $g \ge 2r+3$ if $d=g+r-3$.

math.AG↗

arXiv2019

On the Hilbert scheme of linearly normal curves in $\mathbb{P}^4$ of degree $d = g+1$ and genus $g$

We denote by $\mathcal{H}_{d,g,r}$ the Hilbert scheme of smooth curves, which is the union of components whose general point corresponds to a smooth irreducible and non-degenerate curve of degree $d$ and genus $g$ in $\mathbb{P}^r$. In this article, we show that any non-empty $\mathcal{H}_{g+1,g,4}$ has only one component whose general element is linear normal unless $g=9$. If $g=9$, we show that $\mathcal{H}_{g+1,g,4}$ is reducible with two components and a general element of each component is linearly normal. This establishes the validity of a certain modified version of an assertion of Severi regarding the irreducibility of $\mathcal{H}_{d,g,r}$ for the case $d=g+1$ and $r=4$.

math.AG↗

arXiv2018

Irreducibility and components rigid in moduli of the Hilbert scheme of smooth curves

Denote by $\mathcal{H}_{d,g,r}$ the Hilbert scheme of smooth curves, that is the union of components whose general point corresponds to a smooth irreducible and non-degenerate curve of degree $d$ and genus $g$ in $\mathbb P^r$. A component of $\mathcal{H}_{d,g,r}$ is rigid in moduli if its image under the natural map $π:\mathcal{H}_{d,g,r} \dashrightarrow \mathcal{M}_{g}$ is a one point set. In this note, we provide a proof of the fact that $\mathcal{H}_{d,g,r}$ has no components rigid in moduli for $g > 0$ and $r=3$, from which it follows that the only smooth projective curves embedded in $\mathbb P^3$ whose only deformations are given by projective transformations are the twisted cubic curves. In case $r \geq 4$, we also prove the non-existence of a component of $\mathcal{H}_{d,g,r}$ rigid in moduli in a certain restricted range of $d$, $g>0$ and $r$. In the course of the proofs, we establish the irreducibility of $\mathcal{H}_{d,g,3}$ beyond the range which has been known before.

math.AG↗

arXiv2017

Irreducibility of the Hilbert scheme of smooth curves in $\Bbb P^4$ of degree $g+2$ and genus $g$

We denote by $\mathcal{H}_{d,g,r}$ the Hilbert scheme of smooth curves, which is the union of components whose general point corresponds to a smooth irreducible and non-degenerate curve of degree $d$ and genus $g$ in $\PP^r$. In this note, we show that any non-empty $\mathcal{H}_{g+2,g,4}$ is irreducible without any restriction on the genus $g$. Our result augments the irreducibility result obtained earlier by Hristo Iliev(2006), in which several low genus $g\le 10$ cases have been left untreated.

math.AG↗

arXiv1995

On triple coverings of irrational curves

Given a triple covering $X$ of genus $g$ of a general (in the sense of Brill-Noether) curve $C$ of genus $h$, we show the existence of base-point-free pencils of degree $d$ which are not composed with the triple covering for any $d\ge g-[{3h+1\over 2}]-1$ by utilizing some enumerative methods and computations. We also discuss about the sharpness of our main result and the so-called Castelnuovo-Severi bound byexhibiting some examples.

alg-geom↗