Existence and the reducibility of the Hilbert scheme of linearly normal\n curves in \ℙ^r of relatively high degrees

2021/01/02 by Changho Keem, Keem, Changho
Arts and Humanities · Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Meromorphic and Entire Functions #Primary 14C05 #Religion, Gender, and Enlightenment #Secondary 14H10

paper · pdf · doi:10.48550/arxiv.2101.00559

openalex publication_date 2021/01/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \H_d,g,r be the Hilbert scheme parametrizing smooth\nirreducible and non-degenerate curves of degree d and genus g in\n\ℙ^r. We denote by \H^\L_d,g,r the union of\nthose components of \H_d,g,r whose general element is linearly\nnormal. In this article we show that \H^\L_d,g,r (d\≥\ng+r-3) is non-empty in a certain optimal range of triples (d,g,r) and is\nempty outside the range. This settles the existence (or non-emptiness if one\nprefers) of the Hilbert scheme \H^\L_d,g,r of linearly\nnormal curves of degree d and genus g in \ℙ^r for g+r-3\≤ d\≤\ng+r, r\≥ 3. We also determine all the triples (d,g,r) with g+r-3\≤ d\≤\ng+r for which \H^\L_d,g,r is reducible (or irreducible).\n

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