Explicit complete curves in the moduli space of curves of genus three
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''.
scientific article; zbMATH DE number 806255
scientific article