COMMENTS ON VALUATIONS ASSOCIATED TO SYSTEMS OF
VERTICES/EDGES AND THE MAIN THEOREM OF POP-STIX

Shinichi Mochizuki
Updated May 2, 2011

Let k be an arbitrary complete discrete valuation ﬁeld of mixed characteristic
defwhose residue characteristic we denote by p, k an algebraic closure of k, Gk =Gal(k/k), Σ a set of primes that contains a prime l = p, X a proper hyperbolic
curve over k. Suppose, further, that k is l-cyclotomically full, i.e., that the image
×of the l-adic cyclotomic character Gk → Z×l is open in Zl . Write(Σ)ΠX  ΠX

for the geometrically pro-Σ quotient of the étale fundamental group ΠX of X. Thus,(Σ)
we have a natural surjection ΠX  Gk . Let. . . → Xi+1 → Xi → . . .
[where i ranges over the positive integers] be a coﬁnal system of ﬁnite étale connected(Σ)
Galois coverings of X with stable reduction arising from open subgroups of ΠXand(Σ)
s : Gk → ΠX
(Σ)

a section of ΠX  Gk . Then in the “Comments on a Combinatorial Version ofthe Section Conjecture and the Main Theorem of Pop-Stix” dated March 3, 2011(cf. [CbSC], (5)), we showed that(∗v/e ) [after possibly passing to a coﬁnal subsystem of the given system ofcoverings] there exists either a [not necessarily unique] system of vertices. . .  vi+1  vi  . . .
or a [not necessarily unique] system of edges. . .  ei+1  ei  . . .
— i.e., each vi (respectively, ei ) is an irreducible component (respectively,node) of the special ﬁber of the stable model Xi of Xi that is ﬁxed by
the natural action of the image Im(s) of the section s; the image of theTypeset by AMS-TEX1

2SHINICHI MOCHIZUKI

irreducible component vi+1 (respectively, node ei+1 ) in Xi is contained in
the irreducible component vi (respectively, node ei ).
In the present note, we verify (cf. (1), (2) below), by means of a quite elementaryargument in scheme theory/commutative algebra, that(∗val ) such a system of vertices or edges determines a system of valuations of
the function ﬁelds Ki of the Xi that are ﬁxed by the natural action ofIm(s).
In particular, we obtain a proof of the main theorem of Pop-Stix (cf. [PS]) by meansof elementary graph-theoretic and scheme-/ring-theoretic considerations, without
resorting to the use of highly nontrivial arithmetic results such as Tamagawa’s“resolution of nonsingularities” [i.e., the main result of [Tama]]. Here, we recall that
this result of [Tama] depends, in an essential way, on highly arithmetic argumentsthat require one to take Σ to be the set of all primes, as well as on relatively
deep wild ramiﬁcation properties of p-power coverings of X. In particular, theessential role played by this result in the proof of [PS] has the eﬀect of portraying
the phenomenon discussed in the main theorem of [PS] as being a consequence ofsuch deep arithmetic considerations. In fact, however, the arguments of the present
note imply thatthe essential phenomenon discussed in the main theorem of [PS] is [not“arithmetic” or “p-adic”, but rather] “l-adic” and “combinatorial” in nature and may be obtained as a consequence of quite elementary considerations concerning ﬁnite group actions on graphs and scheme theory/commutative algebra.

(1) Suppose that one has a system of vertices {vi } as in (∗v/e ). If [after possiblypassing to a coﬁnal subsystem of the given system of coverings] each vi+1 maps
quasi-ﬁnitely to vi , then the system of valuations associated to the vi already yields asystem of valuations as desired. Thus, [after possibly passing to a coﬁnal subsystemof the given system of coverings] we may assume without loss of generality that vi+1maps to a closed point xi of vi . If [after possibly passing to a coﬁnal subsystemof the given system of coverings] the xi are all nodes, then we obtain a system ofedges {ei } as in (∗v/e ); this situation will be dealt with in (2) below. Thus, [afterpossibly passing to a coﬁnal subsystem of the given system of coverings] we may
assume without loss of generality that each xi is a smooth point. In particular, thelocal ring Ri of Xi at xi is regular of dimension 2, hence a UFD. Writeordi : Ki× → Q
for the valuation associated to vi , normalized so as to restrict to a ﬁxed [i.e., independent of i], given valuation on k. Then it follows immediately from the deﬁnitionof xi , together with the fact that Ri is a UFD, that we haveordj  (f ) ≥ ordj (f ) ≥ 0

VALUATIONS AND THE MAIN THEOREM OF POP-STIX

3for any nonzero f ∈ Ri ⊆ Ki , j  ≥ j ≥ i. [Here, we think of the various Ki as
being related to one another via the natural inclusions Ki ⊆ . . . ⊆ Kj ⊆ . . . ⊆ Kj  .]Next, let us observe that it follows immediately from the fact that each ordj (−) isdefa valuation that, if we set ordj (0) = +∞, then the subset
defRi ⊇ Ii = {f ∈ Ri | lim ordj (f ) = +∞}j→∞is, in fact, a prime ideal of Ri whose intersection with the ring of integers Ok ⊆ Ri ofk is equal to {0}. In particular, the height of Ii is ≤ 1. If [after possibly passing to a
coﬁnal subsystem of the given system of coverings] the Ii are all of height 1, then it
follows immediately that Ii determines a closed point ξi of Xi , and that the systemof valuations associated to the ξi yields a system of valuations as desired [indeed, ofthe “ideal type”, from the point of view of the original Section Conjecture!]. Thus,[after possibly passing to a coﬁnal subsystem of the given system of coverings] wemay assume without loss of generality that each Ii is of height 0, hence equal to{0}. But this implies that, for f ∈ Ki× , the quantity
def

ord∞ (f ) =lim ordj (f ) ∈ Rj→∞is well-deﬁned. Moreover, one veriﬁes immediately that ord∞ (−) determines avaluation on Ki that is ﬁxed by the action of Im(s). In particular, one obtains asystem of valuations as desired.
(2) Suppose that one has a system of edges {ei } as in (∗v/e ). Write Xilog for theregular log scheme whose underlying scheme is X and whose interior is the genericﬁber Xi ⊆ Xi . Thus, the characteristic of the log structure of Xilog at xi determines— by tensoring the groupiﬁcation of the characteristic with R — a 2-dimensional
real vector space, whose dual we denote by Mi . Thus, Mi is equipped with anatural positive rational structure Pi [i.e., a submonoid isomorphic to Q≥0 ⊕ Q≥0that generates Mi as a real vector space]. [Put another way, Mi is the sort of realvector space that appears in discussions of toric varieties.] The natural morphismlog
Xi+1→ Xilog induces an R-linear map of vector spaces Mi+1 → Mi of rank ≥ 1that maps Pi+1 into Pi . Write P i ⊆ Mi for the closure of Pi in Mi . Let us referto as a P -ray of Mi a ray of Mi emanating from the origin that is contained in P i .
Now it follows immediately from the compactness of the space of P -rays of Mi that[after possibly passing to a coﬁnal subsystem of the given system of coverings] wemay assume that there exists a compatible system {λi } of P -rays of the Mi whichare, moreover, ﬁxed by the action of Im(s). Suppose that [after possibly passing
to a coﬁnal subsystem of the given system of coverings] each λi is rational [i.e.,generated by an element of Pi ]. Then λi corresponds to an irreducible component
vi of a suitable blow-up of Xi at ei ; one may construct these blow-ups so that vi+1
maps into vi . If [after possibly passing to a coﬁnal subsystem of the given systemof coverings] each vi+1 maps quasi-ﬁnitely to vi , then the system of valuations
associated to the vi already yields a system of valuations as desired. Thus, [afterpossibly passing to a coﬁnal subsystem of the given system of coverings] we may
assume without loss of generality that vi+1 maps to a closed point xi of vi ; moreover,
it follows immediately from the fact that the λi form a compatible system that each4SHINICHI MOCHIZUKI

xi is a smooth point. Thus, one may construct either a system of closed points ξiof Xi or a system of “limit valuations ord∞ (−)” as in (1); this yields a systemof valuations as desired. This completes the proof in the case where the λi arerational. Thus, [after possibly passing to a coﬁnal subsystem of the given system ofcoverings] we may assume without loss of generality that each λi is irrational. But
then it is well-known that each λi determines a valuation on Ki ; the compatibilityof these valuations as one varies i follows immediately from the compatibility of theλi . Thus, one obtains a system of valuations as desired.
(3) The present note beneﬁted from discussions with Fumiharu Kato in November
2010.

Bibliography[CbSC] S. Mochizuki, Comments on a Combinatorial Version of the Section Conjecture
and the Main Theorem of Pop-Stix, manuscript dated March 3, 2011.
[PS] F. Pop and J. Stix, Arithmetic in the fundamental group of a p-adic curve —On the p-adic section conjecture for curves, preprint, Philadelphia-HeidelbergCambridge, August 2010.
[Tama] A. Tamagawa, Resolution of nonsingularities of families of curves, Publ. Res.
Inst. Math. Sci. 40 (2004), pp. 1291-1336.