TY - JOUR

T1 - MR 3215343 Reviewed Pirola G.P., Rizzi C. and Schlesinger E. A new curve algebraically but not rationally uniformized by radicals. Asian J. Math., 18 (2014), 127–142. (Reviewer Francesca Vetro) 14H30 (14H10)

AU - Vetro, Francesca

PY - 2014

Y1 - 2014

N2 - A smooth projective complex curve C is called rationally uniformized by radicals if there exists a covering map C \rightarrow P^1 with solvable Galois group. C is called algebraically uniformized by radicals if there exists a finite covering C^{\prime} \rightarrow C with C^{\prime} rationally uniformized by radicals. Abramovich and Harris posed the following problem in [D. Abramovich and J. Harris, Abelian varieties and curves in $W_{d}(C)$, Compositio Math., 78 (1991), pp. 227–-238].\vspace{1ex} Statement S(d, h, g): \textit{Suppose C^{\prime} \rightarrow C is a nonconstant map of smooth curves with C of genus g. If C^{\prime} admits a map of degree d or less to a curve of genus h or less, then so does C. For which values of d, h and g, is true condition S(d, h, g)?In this paper, the authors give new examples of curves algebraically but not rationally uniformized by radicals. They construct such curves of genus 9 in a linear system on the second symmetric product of a curve of genus 2. Furthermore, the curves constructed by the authors are examples of genus 9 curves that does not satisfy condition S(4, 2, 9).

AB - A smooth projective complex curve C is called rationally uniformized by radicals if there exists a covering map C \rightarrow P^1 with solvable Galois group. C is called algebraically uniformized by radicals if there exists a finite covering C^{\prime} \rightarrow C with C^{\prime} rationally uniformized by radicals. Abramovich and Harris posed the following problem in [D. Abramovich and J. Harris, Abelian varieties and curves in $W_{d}(C)$, Compositio Math., 78 (1991), pp. 227–-238].\vspace{1ex} Statement S(d, h, g): \textit{Suppose C^{\prime} \rightarrow C is a nonconstant map of smooth curves with C of genus g. If C^{\prime} admits a map of degree d or less to a curve of genus h or less, then so does C. For which values of d, h and g, is true condition S(d, h, g)?In this paper, the authors give new examples of curves algebraically but not rationally uniformized by radicals. They construct such curves of genus 9 in a linear system on the second symmetric product of a curve of genus 2. Furthermore, the curves constructed by the authors are examples of genus 9 curves that does not satisfy condition S(4, 2, 9).

UR - http://hdl.handle.net/10447/103606

UR - http://www.ams.org

M3 - Review article

VL - 2014

JO - MATHEMATICAL REVIEWS

JF - MATHEMATICAL REVIEWS

SN - 0025-5629

ER -