Studying two point branched Galois covers of the projective line we prove the Inertia Conjecture for the Alternating groups A p+1 , A p+3 , A p+4 for any odd prime p ≡ 2 (mod 3) and for the group A p+5 when additionally 4 ∤ (p + 1) and p ≥ 17. We obtain a generalization of a patching result by Raynaud which reduces these proofs to showing the realizations of the étale Galois covers of the affine line with a fewer candidates for the inertia groups above ∞. We also pose a general question motivated by the Inertia Conjecture and obtain some affirmative results. A special case of this question, which we call the Generalized Purely Wild Inertia Conjecture, is shown to be true for the groups for which the purely wild part of the Inertia Conjecture is already established. In particular, we show that if this generalized conjecture is true for the groups G 1 and G 2 which do not have a common quotient, then the conjecture is also true for the product G 1 × G 2 .
The wild part of Abhyankar's Inertia Conjecture for a product of certain Alternating groups is shown for any algebraically closed field of odd characteristic. For d a multiple of the characteristic of the base field, a newétale A d -cover of the affine line is obtained using an explicit equation and it is shown that it has the minimal possible upper jump.Conjecture 1.1. Let G be a finite quasi p-group.Wild Part: A p-subgroup P of G occurs as the inertia group of a G-Galois cover of P 1 branched only at ∞ if and only if the conjugates of P generate G. Tame Part: Assume that a p-subgroup P of G occurs as the inertia group of a G-Galoisétale cover of the affine line. Let β be an element of G of prime-to-p order contained in the normalizer of P in G. Let I be an extension of β by P in G. Then there is a G-Galois cover φ : Y → P 1 branched only at ∞ with inertia group I at a point in Y over ∞.By Abhyankar's Lemma ([18, XIII, Proposition 5.2]), one can reduce the tame part of the inertia group. On the other hand, Harbater has shown ([6, Theorem 2]) that the wild part of the inertia group can be increased. As a consequence, the inertia conjecture
Let f : X −→ Y be a separable finite surjective map between irreducible normal projective varieties defined over an algebraically closed field, such that the corresponding homomorphism between étale fundamental groups f * :Fix a polarization on Y and equip X with the pullback, by f , of this polarization on Y . Given a stable vector bundle E on X, we prove that there is a vector bundle W on Y with f * W isomorphic to E if and only if the direct image f * E contains a stable vector bundle F such that degree(F ) rank(F ) = 1 degree(f )• degree(E) rank(E) .We also prove that f * V is stable for every stable vector bundle V on Y .
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