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

TL;DR

This paper constructs a family of algebraic curves on cones over certain base curves and demonstrates that this family forms a non-reduced, irreducible component of the Hilbert scheme, revealing new non-reduced components in the moduli space of curves.

Contribution

It introduces a specific family of curves on cones that form non-reduced components of the Hilbert scheme, expanding understanding of its structure.

Findings

Existence of non-reduced components in the Hilbert scheme.

Dimension formulas for the family and deformation spaces.

Construction of curves on cones over base curves with specified genus and degree.

Abstract

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 $\dim T_{[X^{\prime}]} \mathcal{D}^{\prime} = \dim \mathcal{D}^{\prime} + 1 = \dim \mathcal{F}^{\prime} + 1$.