Quotient singularities of products of two curves

Abstract

We give a method to resolve a quotient surface singularity which arises as the quotient of a product action of a finite group on two curves. In the characteristic zero case, the singularity is resolved by means of a continued fraction, which is known as the Hirzebruch-Jung desingularization. We develop the method in the positive characteristic case where the square of the characteristic does not divide the order of the group.Résumé. -Nous donnons une méthode pour résoudre une singularité quotient de surface qui se présente comme le quotient d'une action produit d'un groupe fini sur deux courbes. En caractéristique nulle, la singularité est résolue au moyen d'une fraction continue (désingularisation de Hirzebruch-Jung). Nous développons la méthode dans le cas de la caractéristique strictement positive où le carré de la caractéristique ne divise pas l'ordre du groupe.

Discriminant groups of wild cyclic quotient singularities

“…(a) The cases s = 0 and s = 1 are covered by Theorem 7.1 and Theorem 6.3, respectively. The cases with s ≥ 2 and s ≡ 1 mod p were obtained earlier in the papers [30] and [33].…”

“…Peskin's singularity with µ = 1 introduced above, and all the singularities considered in [30] or [33], are also induced by an action that is ramified precisely at the origin. When p = 2, none of the known explicit resolutions for examples in these classes of singularities produce an associated discriminant group Φ N with order 2 s and s odd.…”

“…In this article, we consider which exponents s ≥ 0 can arise in this way. By studying diagonal actions on products of curves, the first author [30,Theorem 3.15] produced wild quotient singularities with |Φ N | = p s for all exponents s ≥ 2 with s ≡ 1 modulo p. Mitsui [33] later explicitly resolved all wild quotient singularities arising from product of curves, and showed that the previous list is the complete list of exponents arising from product of curves. The missing exponents are then s = 0, as well as all s with s ≡ 1 mod p. 0.1.…”