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

Changho Keem

TL;DR

The paper advances the understanding of irreducibility for the Hilbert schemes $\mathcal{H}_{16,g,5}$ of smooth, nondegenerate curves of degree $16$ in $\mathbb{P}^5$, particularly for genera up to $21$. Using Castelnuovo bounds $\pi(16,5)=21$ and $\pi_1(16,5)=18$, it analyzes curves lying on ambient degree-$4$ surfaces (Veronese, rational normal scrolls, cones) to describe the components and the moduli-fiber structure, obtaining a three-component reducible instance at $g=18$ and detailing behavior for $19\le g\le 21$. The work integrates Severi varieties, $k$-gonal linear series, and moduli-map considerations to bound dimensions and characterize gonality and residual series on components, while also addressing unresolved cases $\mathcal{H}_{16,16,5}$ and $\mathcal{H}_{16,15,5}$ with partial irreducible families. Overall, the paper contributes to the broader Severi-type irreducibility program in $\mathbb{P}^5$ and clarifies how ambient minimal-degree surfaces govern the structure of these Hilbert schemes.

Abstract

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 $\mathcal{H}_{d,g,5}\stackrelμ{\to}\mathcal{M}_g$ and several key properties such as gonality of a general element as well as characterizing smooth elements in each component.

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

TL;DR

The paper advances the understanding of irreducibility for the Hilbert schemes of smooth, nondegenerate curves of degree in , particularly for genera up to . Using Castelnuovo bounds and , it analyzes curves lying on ambient degree- surfaces (Veronese, rational normal scrolls, cones) to describe the components and the moduli-fiber structure, obtaining a three-component reducible instance at and detailing behavior for . The work integrates Severi varieties, -gonal linear series, and moduli-map considerations to bound dimensions and characterize gonality and residual series on components, while also addressing unresolved cases and with partial irreducible families. Overall, the paper contributes to the broader Severi-type irreducibility program in and clarifies how ambient minimal-degree surfaces govern the structure of these Hilbert schemes.

Abstract

We denote by 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 and genus in . In this article, we study for almost every possible genus and chasing after its irreducibility. We also study the natural moduli map and several key properties such as gonality of a general element as well as characterizing smooth elements in each component.

Paper Structure

This paper contains 13 sections, 13 theorems, 34 equations.

Table of Contents

  1. An overview, motivation and preliminary set up
  2. Notations and conventions
  3. Preliminary set up
  4. Organization of the paper
  5. Some relevant generalities and auxiliary results
  6. Severi variety of nodal curves on Hirzebruch surfaces
  7. Linear systems on $k$-gonal curves and Castelnuovo-Severi inequality
  8. On moduli map
  9. Curves of degree $16$ in $\mathbb{P}^5$
  10. Irreducibility, description of general element of $\mathcal{H}_{16,g,5}$ and the moduli map for $19\le g\le 21$
  11. Digression, remarks on $\mathcal{H}_{16,16,5}$ and $\mathcal{H}_{16,15,5}$
  12. Irreducible families in $\mathcal{H}_{16,16,5}$
  13. Irreducible families in $\mathcal{H}_{16,15,5}$

Key Result

Proposition 1.1

For non-negative integers $d$, $g$ and $r$, let be the Brill-Noether number. The dimension of any component of $\mathcal{G}^{r}_{d}$ is at least hence the dimension of any component of $\mathcal{}{H}_{d,g,r}$ is at least Moreover, if $\rho(d,g,r)\ge 0$, there exists a unique component $\mathcal{G}_0\subset\widetilde{\mathcal{G}}$ which dominates $\mathcal{M}$(or $\mathcal{M}_g$).

Theorems & Definitions (27)

  • Proposition 1.1
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Remark 2.8: Castelnuovo-Severi inequality
  • Lemma 2.9
  • ...and 17 more