zbMATH — the first resource for mathematics

Moduli for stable marked trees of projective lines. (English) Zbl 0723.14020
The moduli theory for stable n-pointed trees of projective lines as developed by L. Gerritzen, the author and M. van der Put [Indagationes Math. 50, No.2, 131-163 (1988; Zbl 0698.14019)] is extended to the case of an infinite set M of markings. This is the setting which is relevant for the uniformization theory of stable Riemann surfaces [cf. L. Gerritzen and the author, J. Reine Angew. Math. 389, 190-208 (1988; Zbl 0639.30040)].
Although the final results are formally analogous to the case of a finite set of markings, new technical tools had to be developed even to state the results: since neither the classified objects nor (in general) the moduli spaces are varieties (or schemes), one has to work in the category of provarieties (or projective limits of varieties). We also need new definitions of trees (not relying on edges or on topological properties), of trees of projective lines, and of intersection graphs.
The main result states the existence of fine moduli spaces for stable M- marked trees of projective lines, and for trees where the M marking is equivariant with respect to a given group action.

14H10 Families, moduli of curves (algebraic)
14D22 Fine and coarse moduli spaces
14A20 Generalizations (algebraic spaces, stacks)
Full Text: DOI EuDML
[1] [EGA] Grothendieck, A.: El?ments de g?om?trie alg?brique. IV. Publ. Math. Inst. Hautes Etud. Sci.28, (1966)
[2] [GH] Gerritzen, L., Herrlich, F.: The extended Schottky space. J. Reine Angew. Math.389, 190-208 (1988) · Zbl 0639.30040
[3] [GHP] Gerritzen, L., Herrlich, F., van der Put, M.: Stablen-pointed trees of projective lines. Indag. Math.50, 131-163 (1988) · Zbl 0698.14019
[4] [H] Herrlich, F.: The extended Teichm?ller space. Math. Z.203, 279-291 (1990) · Zbl 0662.32021 · doi:10.1007/BF02570736
[5] [K] Knudsen, F.: The projectivity of the moduli space of stable curves. II. Math. Scand.52, 161-199 (1983) · Zbl 0544.14020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.