arXiv subjects

Seonja Kim

Publications and source records attributed to Seonja Kim.

14 recordsLinked to original sources

arXiv2026

Universal extension spaces and modular maps: unveiling irreducible components of Brill-Noether loci of stable bundles on a general $\nu$-gonal curve

We investigate the Brill-Noether theory of rank-two, degree-$d$ stable vector bundles of speciality $3$ on a general $\nu$-gonal curve of genus $g$, $3 \leq \nu < \lfloor \frac{g+3}{2} \rfloor$. Our approach leverages universal extension spaces, modular maps, and recent advancements in rank-one Brill-Noether theory over Hurwitz spaces. We establish existence criteria for the corresponding Brill-Noether loci and provide a comprehensive description of their irreducible components. We moreover prove that these components exhibit diverse geometric behaviors, categorized by their regularity, superabundance, and the properties of their general points. Notably, for specific degrees $d$, we prove the coexistence of multiple superabundant components alongside a regular one. Using specialization techniques, we uncover a stratification into locally closed subschemes within these components and provide insights into their birational geometry and local structure. Furthermore, our results yield also consequences for Brill-Noether loci of stable, rank-two bundles with a fixed general determinant.

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arXiv2025

Decomposition of the Tschirnhausen module for coverings on decomposable $\mathbb{P}^1$-bundles

In this note, we show that for a smooth algebraic variety $Y$ and a smooth $m$-section $X$ of the $\mathbb{P}^1$-bundle \[ f : \mathbb{P}(\mathcal{O}_Y \oplus \mathcal{O}_Y(E)) \longrightarrow Y, \] where $E$ is an effective divisor on $Y$ satisfying $H^1(Y, \mathcal{O}_Y(kE)) = 0$ for all $k = 1, \ldots, m-1$, the Tschirnhausen module of the induced covering $ f|_X : X \longrightarrow Y $ is completely decomposable. We then apply it to coverings of curves arising in such a way.

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arXiv2025

On the Tschirnhausen module of coverings of curves on decomposable ruled surfaces and applications

We show that for two classes of $m$-secant curves $X \subset S$, with $m \geq 2$, where $f : S = \mathbb{P} (\mathcal{O}_Y \oplus \mathcal{O}_Y (E)) \to Y$ and $E$ is a non-special divisor on a smooth curve $Y$, the Tschirnhausen module $\mathcal{E}^{\vee}$ of the covering $\varphi = f_{|_X} : X \to Y$ decomposes completely as a direct sum of line bundles. Specifically, we prove that: for $X \in |\mathcal{O}_S (mH)|$, where $H$ denotes the tautological divisor on $S$, one has $ \mathcal{E}^{\vee} \cong \mathcal{O}_Y (-E) \oplus \cdots \oplus \mathcal{O}_Y (-(m-1)E) $; for $X \in |\mathcal{O}_S (mH + f^{\ast}q))|$, where $q$ is a point on $Y$, $ \mathcal{E}^{\vee} \cong \mathcal{O}_Y (-E-q) \oplus \cdots \oplus \mathcal{O}_Y (-(m-1)E-q) $ holds. This decomposition enables us to compute the dimension of the space of global sections of the normal bundle of the embedding $X \subset \mathbb{P}^R$ induced by the tautological line bundle $|\mathcal{O}_S (H)|$, where $R = \dim |\mathcal{O}_S (H)|$. As an application, we construct new families of generically smooth components of the Hilbert scheme of curves, including components whose general points correspond to non-linearly normal curves, as well as nonreduced components.

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arXiv2025

Complete classification of irreducible components of the Brill-Noether locus of rank-$2$ vector bundles of degree $d$ and speciality $2$ on a general $\nu$-gonal curve

This paper replaces the previous longer version and focuses on the specialty $2$ case. More precisely, in this paper we address the Brill-Noether theory for rank-two, degree $d$ stable bundles of speciality $2$ on a general $\nu$-gonal curve $C$ of genus $g$, $3 \leq \nu < \lfloor \frac{g+3}{2}\rfloor$, leveraging universal extension spaces, modular maps and recent developments in rank-one Brill-Noether theory over Hurwitz spaces on $C$. We completely classify the irreducible components of such Brill-Noether loci in the whole range of interest for $d$, namely $2g-2 \leq d \leq 4g-4$. Using specialization techniques, we further uncover a stratification into locally closed subsets within some of these components, and we also provide additional insight into the birational geometry and the local structure of every such a component. Our methods yield descriptions of the irreducible components of any such Brill-Noether locus and, as a by-product of our more general results, also derive interesting consequences for Brill-Noether loci of stable, rank-two bundles with a fixed general determinant, rather than fixed degree.

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arXiv2023

Non-reduced components of the Hilbert scheme of curves using triple covers

In this paper we consider curves on a cone that pass through the vertex and are also triple covers of the base of the cone, which is a general smooth curve of genus $\gamma$ and degree $e$ in $\mathbb{P}^{e-\gamma}$. Using the free resolution of the ideal of such a curve found by Catalisano and Gimigliano, and a technique concerning deformations of curves introduced by Ciliberto, we show that the deformations of such curves remain on cones over a deformation of the base curve. This allows us to prove that for $\gamma \geq 3$ and $e \geq 4\gamma + 5$ there exists a non-reduced component $\mathcal{H}$ of the Hilbert scheme of smooth curves of genus $3e + 3\gamma$ and degree $3e+1$ in $\mathbb{P}^{e-\gamma+1}$. We show that $\dim T_{[X]} \mathcal{H} = \dim \mathcal{H} + 1 = (e - \gamma + 1)^2 + 7e + 5$ for a general point $[X] \in \mathcal{H}$.

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arXiv2022

Components of the Hilbert Scheme of smooth projective curves using ruled surfaces II: existence of non-reduced components

For $\gamma \geq 7$ and $g \geq 6\gamma + 5$, we construct a family $\mathcal{F}^{\prime}$ of curves lying on cones in $\mathbb{P}^{g-3\gamma+1}$ over smooth non-degenerate curves of genus $\gamma$ and degree $g-2\gamma$ in $\mathbb{P}^{g-3\gamma+1}$. We show that $\dim \mathcal{F}^{\prime} = 2g-\gamma-1 + (g-3\gamma+1)^2$. For a general curve $X^{\prime}$ from the family $\mathcal{F}^{\prime}$, we compute the dimension of the space of its first-order deformations. We prove that the family $\mathcal{F}^{\prime}$ gives rise to an irreducible, non-reduced component $\mathcal{D}^{\prime}$ of the Hilbert scheme $\mathcal{I}_{2g-4\gamma + 1, g, g - 3\gamma + 1}$, which parametrizes smooth, irreducible, non-degenerate curves of degree $2g-4\gamma + 1$ and genus $g$ in $\mathbb{P}^{g-3\gamma+1}$. We obtain $\dim T_{[X^{\prime}]} \mathcal{D}^{\prime} = \dim \mathcal{D}^{\prime} + 1 = \dim \mathcal{F}^{\prime} + 1$.

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arXiv2020

Linear series on a curve of compact type bridged by a chain of elliptic curves

In the present paper we investigate conditions for the non-existence of a limit linear series on a curve of compact type such that two smooth curves are bridged by a chain of two elliptic curves. Combining this work with results on the existence of a smoothable limit linear series on such a curve, we show relations among Brill-Noether loci of codimension at most two in the moduli space of complex curves. Specifically, Brill-Noether loci of codimension two have mutually distinct supports.

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arXiv2020

Components of the Hilbert Scheme of smooth projective curves using ruled surfaces

Let $\mathcal{I}_{d,g,r}$ be the union of irreducible components of the Hilbert scheme whose general points correspond to smooth irreducible non-degenerate curves of degree $d$ and genus $g$ in $\mathbb{P}^r$. We use families of curves on cones to show that under certain numerical assumptions for $d$, $g$ and $r$, the scheme $\mathcal{I}_{d,g,r}$ acquires generically smooth components whose general points correspond to curves that are double covers of irrational curves. In particular, in the case $ρ(d,g,r) := g-(r+1)(g-d+r) \geq 0$ we construct explicitly a regular component that is different from the distinguished component of $\mathcal{I}_{d,g,r}$ dominating the moduli space $\mathcal{M}_g$.

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arXiv2018

Moduli spaces of bundles and Hilbert schemes of scrolls over $ν$-gonal curves

The aim of this paper is two--fold. We first strongly improve our previous main result Theorem 3.1 in Arxiv 1702.00918v3 12Feb2018 ("Brill-Noether loci of rank two vector bundles on a general $ν$-gonal curve"), concerning classification of irreducible components of the Brill--Noether locus parametrizing rank 2 semistable vector bundles of suitable degrees $d$, with at least $d-2g+4$ independent global sections, on a general $ν$--gonal curve $C$ of genus $g$. We then uses this classification to study several properties of the Hilbert scheme of suitable surface scrolls in projective space, which turn out to be special and stable.

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arXiv2018

Brill-Noether loci of rank two vector bundles on a general $ν$-gonal curve

In this paper we study the Brill Noether locus of rank 2, (semi)stable vector bundles with at least two sections and of suitable degrees on a general $ν$-gonal curve. We classify its reduced components whose dimensions are at least the corresponding Brill-Noether number. We moreover describe the general member $\mathcal F$ of such components just in terms of extensions of line bundles with suitable {\em minimality properties}, providing information on the birational geometry of such components as well as on the very-ampleness of $\mathcal F$.

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arXiv2012

Explicit presentations of nonspecial line bundles and secant spaces

A line bundle L on a smooth curve X is nonspecial if and only if L admits a presentation L=K_X -D +E for some effective divisors D and E>0 on X with gcd (D, E)=0 and h^0 (X, O_X (D))=1. In this work, we define a minimal presentation of L which is minimal with respect to the degree of E among the presentations. If L=K_X -D +E with degE>2 is a minimal, then L is very ample and any q-points of X with q <degE are embedded in general position but the points of E are not. We investigate sufficient conditions on divisors D and E for L=K_X -D +E to be minimal. Through this, for a number n in some range, it is possible to construct a nonspecial very ample line bundle L=K_X -D +E on X with/without an n-secant (n-2)-plane of the embedded curve by taking divisors D and E on X. As its applications, we construct nonspecial line bundles which show the sharpness of Green and Lazarsfeld's Conjecture on property (N_p) for general n-gonal curves and simple multiple coverings of smooth plane curves.

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arXiv2008

Normal generation of line bundles on multiple coverings

Any line bundle $\cl $ on a smooth curve $C$ of genus $g$ with $°\cl \ge 2g+1$ is normally generated, i.e., $φ_\cl (C)\subseteq \mathbb P H^0 (C,\cl)$ is projectively normal. However, it has known that more various line bundles of degree $d$ failing to be normally generated appear on multiple coverings of genus $g$ as $d$ becomes smaller than $2g+1$. Thus, investigating the normal generation of line bundles on multiple coverings can be an effective approach to the normal generation. In this paper, we obtain conditions for line bundles on multiple coverings being normally generated or not, respectively.

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arXiv2006

Normal generation and Clifford index

Let $C$ be a smooth curve of genus $g\ge 4$ and Clifford index $c$. In this paper, we prove that if $C$ is neither hyperelliptic nor bielliptic with $g\ge 2c+5$ and $\mathcal M$ computes the Clifford index of $C$, then either $°\mathcal M\le \frac{3c}{2}+3$ or $|\mathcal M|=|g^1_{c+2}+h^1_{c+2}|$ and $g=2c+5$. This strengthens the Coppens and Martens' theorem (\cite{CM}, Corollary 3.2.5). Furthermore, for the latter case (1) $\mathcal M$ is half-canonical unless $C$ is a $\frac{c+2}{2}$-fold covering of an elliptic curve, (2) $\mathcal M(F)$ fails to be normally generated with $\cli(\mathcal M(F))=c$, $h^1(\mathcal M(F))=2$ for $F\in g^1_{c+2}$. Such pairs $(C,\mathcal M)$ can be found on a $K3$-surface whose Picard group is generated by a hyperplane section in $\mathbb P^r$. For such a $(C, \mathcal M)$ on a K3-surface, $\mathcal M$ is normally generated while $\mathcal M(F)$ fails to be normally generated with $\cli(\mathcal M)=\cli(\mathcal M(F))=c$.

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arXiv2006

Projective Normality Of Algebraic Curves And Its Application To Surfaces

Let $L$ be a very ample line bundle on a smooth curve $C$ of genus $g$ with $\frac{3g+3}{2}<°L\le 2g-5$. Then $L$ is normally generated if $°L>\max\{2g+2-4h^1(C,L), 2g-\frac{g-1}{6}-2h^1(C,L)\}$. Let $C$ be a triple covering of genus $p$ curve $C'$ with $C\stackrelϕ\to C'$ and $D$ a divisor on $C'$ with $4p<°D< \frac{g-1}{6}-2p$. Then $K_C(-ϕ^*D)$ becomes a very ample line bundle which is normally generated. As an application, we characterize some smooth projective surfaces.

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