Asian Journal of Mathematics

Volume 18 (2014)

Number 1

A new curve algebraically but not rationally uniformized by radicals

Pages: 127 – 142



Gian Pietro Pirola (Department of Mathematics, University of Pavia, Italy)

Cecilia Rizzi (Dipartimento di Matematica “F. Brioschi”, Politecnico di Milano, Italy)

Enrico Schlesinger (Dipartimento di Matematica “F. Brioschi”, Politecnico di Milano, Italy)


We give a new example of a curve $C$ algebraically, but not rationally, uniformized by radicals. This means that $C$ has no map onto $\mathbb{P}^1$ with solvable Galois group, while there exists a curve $C'$ that maps onto $C$ and has a finite morphism to $\mathbb{P}^1$ with solvable Galois group. We construct such a curve $C$ of genus $9$ in the second symmetric product of a general curve of genus $2$. It is also an example of a genus $9$ curve that does not satisfy condition $S(4, 2, 9)$ of Abramovich and Harris.


monodromy groups, Galois groups, projective curves

2010 Mathematics Subject Classification

14H10, 14H30, 20B25

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