Geodesic
Representation varieties of the~fundamental groups of non-orientable surfaces
Sbornik. Mathematics, Tome 188 (1997) no. 7, pp. 997-1039

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Let $\Gamma_g$ be the fundamental group of a compact non-orientable surface of genus $g$ and let $K$ be an algebraically closed field of characteristic 0. The structure of the representation varieties $R(\Gamma_g,\mathrm{GL}_n(K))$, $R(\Gamma_g,\mathrm{SL}_n(K))$ of $\Gamma_g$ into $\mathrm{GL}_n(K)$ and $\mathrm{SL}_n(K)$ and of the character varieties $X(\Gamma_g,\mathrm{GL}_n(K))$ is described; namely, the number of their irreducible components and their dimensions are determined and their birational properties are investigated. It is proved, in particular, that all the irreducible components of $R(\Gamma_g,\mathrm{GL}_n(K))$ are $\mathbb Q$-rational varieties. 
@article{SM_1997_188_7_a2,
     author = {V. V. Benyash-Krivets and V. I. Chernousov},
     title = {Representation varieties of the~fundamental groups of non-orientable surfaces},
     journal = {Sbornik. Mathematics},
     pages = {997--1039},
     publisher = {mathdoc},
     volume = {188},
     number = {7},
     year = {1997},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1997_188_7_a2/}
}
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V. V. Benyash-Krivets; V. I. Chernousov. Representation varieties of the~fundamental groups of non-orientable surfaces. Sbornik. Mathematics, Tome 188 (1997) no. 7, pp. 997-1039. http://geodesic.mathdoc.fr/item/SM_1997_188_7_a2/