Explicit complete curves in the moduli space of curves of genus three
type: Item
instance of: scholarly article
title: Explicit complete curves in the moduli space of curves of genus three
zbMATH Open document ID: 0863.14017
DOI: 10.1007/BF01267642
author: Chris Zaal
published in: Geometriae Dedicata
publication date: 1997-06-10T00:00:00Z
review text: It is known that the moduli space \({\mathcal M}_g\) of smooth curves of genus \(g\) contains complete curves if \(g\geq 3\). \textit{J. Harris} [in: Algebraic geometry Proc. Summer Res. Inst., Brunswick 1985; Part I, Proc. Symp. Pure Math. 46, 99-143 (1987; Zbl 0646.14019)] noticed that no explicit examples of these curves were known. Explicit examples have been provided by \textit{G. González Díez} and \textit{W. J. Harvey} [Math. Proc. Camb. Philos. Soc. 110, No. 3, 461-466; 467-472 (1991; Zbl 0761.14007)] when \(g\geq 4\). In the paper under review the author produces such examples for \(g=3\). Briefly the construction is as follows. Let \(C\) be a bielliptic curve of genus 3 and fix a complete curve \(F\subset C\times C\backslash\Delta\). Let \(f\in F\) and denote by \(\pi_f:C_f\to C\) the corresponding double cover of \(C\) branched over \(f\). To the covers \(\pi_f\) one associates the Prym varieties to get a complete family of principally polarized abelian varieties, which turn out to be Jacobians of smooth nonhyperelliptic genus 3 curves. The corresponding family of curves is then produced via the ``trigonal construction''.
Mathematics Subject Classification ID: 14H10
Mathematics Subject Classification ID: 14H45
zbMATH DE Number: 806255
zbMATH Keywords: moduli space
zbMATH Keywords: complete curves
reviewed by: A. Del Centina
MaRDI profile type: Publication
cites work: A bound on the dimensions of complete subvarieties of \({\mathcal M}_ g\).
cites work: The fibers of the Prym map
cites work: On complete curves in moduli space I
cites work: Q4195061
cites work: Q3789680
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description: scientific article; zbMATH DE number 806255