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RESEARCH PRODUCT

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)

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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).