The Oort Conjecture on lifting covers of curves

Abstract

We show that the conjecture of Oort on lifting covers of curves is true. The main ingredients in the proof are a deformation argument in characteristic p and (a special case of) a very recent result by Obus-Wewers. A kind of boundedness result is given as well.

A generalization of the Oort conjecture

“…1.3 Some history leading to Conjecture 1.9 The first major result on the local lifting problem was the 1989 paper [SOS89] of Sekiguchi-Oort-Suwa, which showed that Z/pm is a local Oort group when p ∤ m. That Z/p 2 m is a local Oort group was proven in 1998 by ). The full Oort conjecture was proven in 2014 by and [Pop14]).…”

Global Oort groups

“…If G = A 4 , then every proper subgroup of G is either cyclic or a Klein four group. But these subgroups are Oort groups by [21] and [20] respectively, and hence are local Oort groups as above. So G is an Oort group by [7,Theorem 2.4].…”

“…More recently, in [17], it was shown that a cyclic group G is an Oort group provided that its order is divisible at most by p 3 , or more generally if its higher ramification groups satisfy a certain condition. Finally, by deforming a general cover to one in the special case treated in [17], the full conjecture was proven in [21].…”

The (local) lifting problem for curves

“…Thus δ T /S = 3(ν − 5) + 18 = δ N/L and we are done. To conclude, we remark that [Pop14, Proposition 4.7] and its setup carry through exactly in our situation, with our A N playing the role of A |ι| in [Pop14]. Indeed, we have that the analog of Σ ι in that proposition contains all closed points, by the paragraph above.…”

“…So the local-global principle tells us that Y η → W η lifts to a cover Y O 1 → W O 1 over some characteristic zero complete discrete valuation ring O 1 with residue field k((̟)). Then,[Pop14, Lemma 4.3] shows that we can "glue" the covers Y → W and Y O 1 → W O 1 along the generic fiber of the former and the special fiber of the latter, in order to get a cover defined over a rank two characteristic zero valuation ring O with residue field k lifting Y → W (cf. [Pop14, p. 319, second paragraph]).…”