Wildly ramified actions and surfaces of general type arising from Artin–Schreier curves

Abstract

We analyse the diagonal quotient for the product of certain Artin-Schreier curves. The smooth models are almost always surfaces of general type, with Chern slopes tending asymptotically to 1. The calculation of numerical invariants relies on a close examination of the relevant wild quotient singularity in characteristic p. It turns out that the canonical model has q − 1 rational double points of type A q−1 , and embeds as a divisor of degree q in P 3 , which is in some sense reminiscent of the classical Kummer quartic.

Quotient singularities of products of two curves

“…In this paper, we give an explicit desingularization and calculate the intersection matrix and invariants by means of plural continued fractions in the case where the square of the characteristic does not divide the order of the group. Our result generalizes those in [6] and [10] as explained below. A special case of our result answers the question on the intersection matrix in [10, §1].…”

“…When G ∼ = Z/pZ, our result in the local setting generalizes the previously known results in the global setting in the case α 1 = 1 or α 2 = 1 [10] and in the case α 1 = α 2 = p − 1 [6]. Our approach is different from those in [10] and [6].…”

“…The difficulty derives from that of the calculation of the dimension of the differential forms on the product of two curves invariant under the product action of G (Lemma 8.3). In this section, we calculate p g in the case G ∼ = Z/pZ by generalizing the method in [6]. hold.…”

“…The intersection matrix of the exceptional locus is calculated under certain conditions of the global geometry which are essentially used. The latter [6] uses an explicit defining equation of the invariant ring of the action on the product of two curves.…”

Wild Singularities of Kummer Varieties

“…Unfortunately, the relations between the generators are unknown for p = 2 and n ≥ 3. In the case n = 2 one gets the singularity from [6], Proposition 2.2. Question 5.3.…”