Liftings of coverings of weakly ramified curves. (Relèvements des revêtements de courbes faiblement ramifiés.) (French) Zbl 1108.14024
Let \(X\) be a smooth projective curve over a perfect field \(k\) of characteristic \(p>0\) with a finite group \(G \subset \text{Aut}_k(X)\). Let \(R(X,G)\) be the versal equivariant deformation ring of \((X,G)\). The authors prove the following result:
Theorem. Let us assume that the action of \(G\) is weakly ramified (i.e. for all \(P \in X\) with uniformizer \(x\), \(G_{P,2}:=\{\sigma \in G, \text{ord}_x(\sigma x-x)>2\}\) is trivial). Then one of the following properties is satisfied :
1) \(R(X,G)\) is cancelled by \(p\).
2) the non trivial ramification groups of \(G\) whose order is divisible by \(p\) are either
a) \(\mathbb{Z}/p\mathbb{Z}\) ;
b) the dihedral group \(\mathbb{Z}/p\mathbb{Z} \rtimes \mathbb{Z}/2\mathbb{Z} \) if \(p>2\) or \((\mathbb{Z}/2\mathbb{Z})^2\) if \(p=2\) ;
c) \(A_4=(\mathbb{Z}/2\mathbb{Z})^2 \rtimes \mathbb{Z}/3\mathbb{Z}\) for \(p=2\).
In any of these cases \(R(X,G)\) is a relative complete intersection over the ring of Witt vectors \(W(k)\); in particular \(R(X,G)\) is flat over \(W(k)\).
Thus the characteristic of \(R(X,G)\) is either \(0\) or \(p\). As the authors point out, if the curve \(X\) is ordinary then the action of \(G\) is weakly ramified [S. Nakajima, Trans. Am. Math. Soc. 303, 595–607 (1987; Zbl 0644.14010)].
Theorem. Let us assume that the action of \(G\) is weakly ramified (i.e. for all \(P \in X\) with uniformizer \(x\), \(G_{P,2}:=\{\sigma \in G, \text{ord}_x(\sigma x-x)>2\}\) is trivial). Then one of the following properties is satisfied :
1) \(R(X,G)\) is cancelled by \(p\).
2) the non trivial ramification groups of \(G\) whose order is divisible by \(p\) are either
a) \(\mathbb{Z}/p\mathbb{Z}\) ;
b) the dihedral group \(\mathbb{Z}/p\mathbb{Z} \rtimes \mathbb{Z}/2\mathbb{Z} \) if \(p>2\) or \((\mathbb{Z}/2\mathbb{Z})^2\) if \(p=2\) ;
c) \(A_4=(\mathbb{Z}/2\mathbb{Z})^2 \rtimes \mathbb{Z}/3\mathbb{Z}\) for \(p=2\).
In any of these cases \(R(X,G)\) is a relative complete intersection over the ring of Witt vectors \(W(k)\); in particular \(R(X,G)\) is flat over \(W(k)\).
Thus the characteristic of \(R(X,G)\) is either \(0\) or \(p\). As the authors point out, if the curve \(X\) is ordinary then the action of \(G\) is weakly ramified [S. Nakajima, Trans. Am. Math. Soc. 303, 595–607 (1987; Zbl 0644.14010)].
Reviewer: Christophe Ritzenthaler (Marseille)
MSC:
| 14H30 | Coverings of curves, fundamental group |
| 14H10 | Families, moduli of curves (algebraic) |
| 11G20 | Curves over finite and local fields |
Citations:
Zbl 0644.14010References:
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