Real algebraic curves. (English) Zbl 0533.14011

Gross, Benedict H.; Harris, Joe

Ann. Sci. Éc. Norm. Supér. (4) 14, 157-182 (1981).

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

Keywords:

real theta-characteristics; real abelian varieties; complex conjugation on complex algebraic curves; real curves; Picard scheme; Real hyperelliptic curves; real plane curves; real trigonal curves; real moduli

Citations:

Zbl 0426.14013; Zbl 0492.14015

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