Reducibility of the Hilbert scheme of smooth curves and families of double covers
type: Item
instance of: scholarly article
title: Reducibility of the Hilbert scheme of smooth curves and families of double covers
zbMATH Open document ID: 1390.14019
author: Youngook Choi
author: Seonja Kim
author: Hristo Krassimirov Iliev
published in: Taiwanese Journal of Mathematics
publication date: 2018-05-23T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.twjm/1498874608
review text: Families of curves in projective space have long been in studied in algebraic geometry. In the modern language, let \(H_{d,g} (\mathbb P^r)\) denote the Hilbert scheme parametrizing curves \(C \subset \mathbb P^r\) of degree \(d\) and arithmetic genus \(g\) and let \({\mathcal I}_{d,g,r} \subset H_{d,g} (\mathbb P^r)\) be the union of irreducible components whose general member is a smooth irreducible non-degenerate curve. Almost one hundred years ago \textit{F. Severi} claimed that \({\mathcal I}_{d,g,r}\) is irreducible if \(d \geq g+r\) [Vorlesungen über algebraische Geometrie. Leipzig u. Berlin: B. G. Teubner (1921; JFM 48.0687.01)]. While \textit{L. Ein} showed that the claim is true for \(r=3\) [Ann. Scient. Ec. Norm. Sup. 19, 469--478 (1986; Zbl 0606.14003)] and \(r=4\) [\textit{L. Ein}, Proc. Symp. Pure Math. 46, 83--87 (1987; Zbl 0647.14012)], [\textit{E. Mezzetti} and \textit{G. Sacchiero}, Lect. Notes Math. 1389, 183--194 (1989; Zbl 0716.14012)] and [\textit{C. Keem}, Proc. Am. Math. Soc. 122, No. 2, 349--354 (1994; Zbl 0860.14003)] produced counterexamples for \(r \geq 6\): these examples have been produced by exhibiting families of curves whose general member is an \(m\)-fold cover of \(\mathbb P^1\) for \(m \geq 3\). In the paper under review, the authors produce new irreducible components of \({\mathcal I}_{d,g,r}\) whose general member \(C \subset \mathbb P^r\) has the property that the morphism defined by the linear system \(\mathbb P H^0(C, \omega_C (-1))\) has degree two (possibly after removing base points).
reviewed by: Scott Nollet
Mathematics Subject Classification ID: 14C05
Mathematics Subject Classification ID: 14H10
zbMATH DE Number: 6871333
zbMATH Keywords: Hilbert scheme of smooth connected curves
zbMATH Keywords: Brill-Noether theory
zbMATH Keywords: double covers
MaRDI profile type: Publication
OpenAlex ID: W2618411607
cites work: Topics in the theory of Riemann surfaces
cites work: Q3347989
cites work: Geometry of Algebraic Curves
cites work: Hilbert scheme of smooth space curves
cites work: Q3791255
cites work: Q4749943
cites work: On the Irreducibility of the Hilbert Scheme of Curves inℙ5
cites work: Reducible Hilbert Scheme of Smooth Curves with Positive Brill-Noether Number
cites work: Irreducibility of a subscheme of the Hilbert scheme of complex space curves
cites work: ON THE IRREDUCIBILITY OF THE FAMILY OF ALGEBRAIC CURVES IN COMPLEX PROJECTIVE SPACEPr
cites work: Moduli spaces of algebraic curves with rational maps
cites work: Q3203134
cites work: On the existence of certain families of curves
DOI: 10.11650/TJM/7839
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description: scientific article; zbMATH DE number 6871333