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Group actions, divisors, and plane curves. (English) Zbl 1439.14134

The main objective of this paper “is to provide an elementary introduction to the moduli space of curves of degree \(n\) in the real or complex plane, modulo the action of the group of projective transformations”. The individual sections cover group actions and orbifolds (§2), divisors on the projective line \({\mathbb P}^1\) and the moduli space of effective divisors with finite stabilizer on \({\mathbb P}^1\) (§3), curves (or \(1\)-cycles) in \({\mathbb P}^2\) and their moduli space (§ 4), cubic curves (§5), curves of higher degree (§6), singularity genus and proper action (§7), curves with infinite (§8) and finite automorphism groups (§9), and the Harnack-Hilbert problem for real curves (§10). In an appendix, remarks on the literature are given.

MSC:

14L30 Group actions on varieties or schemes (quotients)
14H50 Plane and space curves
57R18 Topology and geometry of orbifolds
14H10 Families, moduli of curves (algebraic)
08A35 Automorphisms and endomorphisms of algebraic structures
14P25 Topology of real algebraic varieties

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