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

2022/08/26 by Youngook Choi, Choi, Youngook, Hristo Iliev +3 · 1 citation
Computer Science · Mathematics · #14C05 #14H10 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2208.12470

openalex publication_date 2022/08/26 · openalex created_date 2022/08/30 · openalex updated_date 2026/07/28

Abstract

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

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