By Conte A. (ed.)

Lecture notes in arithmetic No.947

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E graph. i-th, JPo,. 2. ,P,}, -+ position; of T on F', point -y the of the of three y consists b) S, is d. = we use the total stable every = a) is not -, b, for a) d(v) b) EvE V d(v) 3. bounds) it unstable edges containing equal to the set of stable = 2. genus is 3. The stabilization Let 1. 4) (pr where the vertical V(P' d; ^ ), 'r, incident with = y(i) i, fe(j) the mark of the is closed a Stacks defines . is the d) -r, fo(j) indexed by a in a has -r, d) element for with xi, we words, maps as in [4].

1. 1, Tpl(-O - - on H'(71, L = f *Tp,(-O dAO. We also oo)(oo). and Tp,(Pl) To calculate - Ao)ioo weights has oo)). - The torus need the of weights these, that note (Ai -Aj)j:Aj. (-00) changes The the by (-0) Twisting But has respectively. Tpi(oo), Tpi(O) weights by Tpi(O) weight AldA has weight and Tpi(oo) oo) are AOdA'. 1 at (oo). A, + uj)it-l Ao (Ai (,))jOo at (0) and (Aj holds same f*. 16) - for ( (1-d). 1. 18) e(g*N) II E V. 3). 18). 12) is Ci2 - replace -ci, for an unstable flag i E F, by the weight of T on where f i, jJ is the edge containing i.

D) (k). Again, morphism M(r') covering followed by 6tale to = the desired obtain is also for in the E V,, v we Pl: = (0, 1), = and done in families C, on oo A) defines This or -+ this d). 2. roots 1. p /-td(v) where Ad(v) i on V(-rs). 'l --+ (4-6) d). 6). 3. 3 Propositions the d) acts Aut(T, d) p the on acts on mor- -H(,rS) immersion V(-r') Putting in Aut (T, d). 7) V(P') V(-r-')IG which is a closed T7 d) / Vg,n (Pr d), (r, d) Aut , immersion. 4. Proposition classes Of Vg,n(]Pr, the image Of 4 (7-,d,-y) d)(k).

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