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

Changho Keem

TL;DR

The paper investigates the Hilbert schemes of smooth projective curves embedded in projective space and their images in the moduli space $\\mathcal{M}_g$, challenging the conjectured minimal dimension bound $X(d,g,r)=3g-3+\\rho(d,g,r)+\\dim\\operatorname{Aut}(\\mathbb{P}^r)$ for components with $\\operatorname{codim}_{\\mathcal{M}_g}\\mu(\\mathcal{H})\\le g-5$. By developing standard Brill-Noether and residual-series frameworks and constructing explicit counterexamples outside the Brill-Noether range, the authors show that the proposed bound can fail; in particular, they exhibit components where the dimension exceeds $X(d,g,r)$ while still satisfying the codimension constraint, thereby disproving the relaxed conjecture with $\\beta(g)=g-5$. They also demonstrate the existence of components with strictly less than the expected number of moduli, providing concrete triples and discussing implications for refined moduli bounds. Overall, the work deepens understanding of the interaction between embedding dimensions, moduli, and the structure of Hilbert schemes, highlighting subtle phenomena beyond classical Brill-Noether expectations.

Abstract

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 $μ: \mathcal{H}_{d,g,r}\dasharrow \mathcal{M}_g$ be the natural map sending $X\in\mathcal{H}_{d,g,r}$ to its isomorphism class $μ(X)=[X]\in\mathcal{M}_g$. It has been conjectured that a component $\mathcal{H}\subset\mathcal{H}_{d,g,r}$ has the minimal possible dimension $$\Xx(d,g,r):=3g-3+ρ(d,g,r)+\dim\operatorname{Aut}(\mathbb{P}^r)$$ \noindent if $\codim_{\mathcal{M}_g}μ(\mathcal{H})\le g-5$ provided $\Xx(d,g,r)\ge 0$, where $ρ(d,g,r):=g-(r+1)(g-d+r)$ is the Brill-Noether number. In this article, we exhibit examples against the conjecture discuss further for the study of the functorial map $μ: \\mathcal{H}{d,g,r}\dasharrow\mathcal{M}_g$ along this line. A component $\mathcal{H}\subset \mathcal{H}_{d,g,r}$ is said to have the {\it expected number of moduli} if $$\dimμ({\Hh})=\min\{3g-3, 3g-3+ρ(d,g,r)\},$$provided $3g-3+ρ(d,g,r)\ge 0$. 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.

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

TL;DR

The paper investigates the Hilbert schemes of smooth projective curves embedded in projective space and their images in the moduli space , challenging the conjectured minimal dimension bound for components with . By developing standard Brill-Noether and residual-series frameworks and constructing explicit counterexamples outside the Brill-Noether range, the authors show that the proposed bound can fail; in particular, they exhibit components where the dimension exceeds while still satisfying the codimension constraint, thereby disproving the relaxed conjecture with . They also demonstrate the existence of components with strictly less than the expected number of moduli, providing concrete triples and discussing implications for refined moduli bounds. Overall, the work deepens understanding of the interaction between embedding dimensions, moduli, and the structure of Hilbert schemes, highlighting subtle phenomena beyond classical Brill-Noether expectations.

Abstract

We denote by the Hilbert scheme of smooth curves of degree and genus in . Denoting by the moduli space of smooth curves of genus , let be the natural map sending to its isomorphism class . It has been conjectured that a component has the minimal possible dimension \noindent if provided , where is the Brill-Noether number. In this article, we exhibit examples against the conjecture discuss further for the study of the functorial map along this line. A component is said to have the {\it expected number of moduli} if provided . 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.
Paper Structure (7 sections, 3 theorems, 64 equations)

This paper contains 7 sections, 3 theorems, 64 equations.

Table of Contents

  1. An overview, preliminaries and basic set-up
  2. Preliminaries, Notations and Some examples
  3. Preliminaries, Notations and Conventions
  4. Some old and new Examples
  5. Examples against Conjecture \ref{['con']} with $\beta(g)=g-5$
  6. Components with less than the expected number of moduli
  7. Hilbert scheme with a component having less than the expected number of Moduli

Key Result

Lemma 2.6

(i) Let $C\stackrel{\eta}{\rightarrow} E$ be a double covering of a curve $E$ of genus $h\ge 1$. Let $\mathcal{}{E}=g^s_{e}$ be a non-special linear series on $E$. Assume that $|\eta^*(g^s_{e})|=g^s_{2e}$. Then the base-point-free part of the complete $|K_C(-\eta^*(g^s_{e}))|$ is compounded. (ii) Le

Theorems & Definitions (21)

  • Conjecture 1.1
  • Example 2.1
  • proof
  • Example 2.4
  • Example 2.5
  • Lemma 2.6
  • proof
  • proof
  • Example 2.7
  • proof
  • ...and 11 more