Non-reduced components of the Hilbert scheme of curves using triple covers

2023/02/17 by Youngook Choi, Choi, Youngook, Hristo Iliev +3 · 2 citations
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.2302.08707

openalex publication_date 2023/02/17 · openalex created_date 2023/02/21 · openalex updated_date 2026/07/28

Abstract

In this paper we consider curves on a cone that pass through the vertex and are also triple covers of the base of the cone, which is a general smooth curve of genus γ and degree e in ℙ^e-γ. Using the free resolution of the ideal of such a curve found by Catalisano and Gimigliano, and a technique concerning deformations of curves introduced by Ciliberto, we show that the deformations of such curves remain on cones over a deformation of the base curve. This allows us to prove that for γ≥ 3 and e ≥ 4γ+ 5 there exists a non-reduced component H of the Hilbert scheme of smooth curves of genus 3e + 3γ and degree 3e+1 in ℙ^e-γ+1. We show that dim T_[X] H = dim H + 1 = (e - γ+ 1)² + 7e + 5 for a general point [X] ∈ H.

Cited by

• Decomposition of the Tschirnhausen module for coverings on decomposable ℙ¹-bundles
• On the Tschirnhausen module of coverings of curves on decomposable ruled surfaces and applications

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