The category of smooth, irreducible, projective, complex algebraic curves is equivalent to the category of compact Riemann surfaces. We study automorphism groups of Riemann surfaces which are equivalent to complex algebraic curves with real moduli. A complex algebraic curve C has real moduli when the corresponding surface \(X_C\) admits an anti-conformal automorphism. If no such an automorphism is an involution (symmetry), then the surface \(X_C\) is called pseudo-real and the curve C is isomorphic to its conjugate, but is not definable over reals. Otherwise, the surface \(X_C\) is called symmetric and the curve C is real.

中文翻译：

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对称和伪实黎曼曲面的自同构群

光滑、不可约、射影、复代数曲线的范畴等价于紧黎曼曲面的范畴。我们研究黎曼曲面的自同构群，它等价于具有实模的复代数曲线。当相应的曲面\(X_C\)承认反共形自同构时，复代数曲线C具有实模量。如果没有这样的自同构是对合（对称），则表面\(X_C\)称为伪实数，曲线C与其共轭同构，但不能在实数上定义。否则，曲面\(X_C\)称为对称曲面，而曲线C为实数。