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Real algebraic curves. (English) Zbl 0533.14011
This is an exhaustive study of the action of the complex conjugation on complex algebraic curves that are defined by real polynomials. The complex conjugation defines an involution of these real curves. This involution defines an action on the symmetric powers of the curve and on the Picard scheme of the curve. The authors study that action and apply it to real theta-characteristics. They show as well how the topological invariants of a real curve X are determined by the action of the complex conjugation on the group \(H_ 1(X({\mathbb{C}}),{\mathbb{Z}}/2)\). This question was also considered by H. Jaffee [Topology 19, 81-87 (1980; Zbl 0426.14013)].
Real hyperelliptic curves, real plane curves and real trigonal curves are considered as examples of the general theory. A topological argument leads to an interesting observation: entire components of the real moduli contain no hyperelliptic curves once the genus is at least 4. The paper ends with remarks on real moduli and with a real form of the Torelli theorem which was also proved independently by R. Silhol [see e.g. Math. Z. 181, 345-364 (1982; Zbl 0492.14015)].
Reviewer: M.Seppälä

MSC:
14H10 Families, moduli of curves (algebraic)
14H25 Arithmetic ground fields for curves
14Pxx Real algebraic and real-analytic geometry
14H40 Jacobians, Prym varieties
14K15 Arithmetic ground fields for abelian varieties
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