Determine asymptotic growth for very general Dolgachev surfaces

Determine whether the degree of irrationality irr(S(p,q)) tends to infinity for a very general Dolgachev surface S(p,q) as the product pq of its relatively prime multiple-fiber multiplicities tends to infinity.

Background

For a Dolgachev surface S(p,q) with relatively prime integers p,q>1, the paper gives the general upper bound irr(S)≤2p2q2. It also proves that a very general surface of type (p,q) has no birational involution and consequently has degree of irrationality at least three.

Special Dolgachev surfaces are constructed with degree two when p=2 and degree three when p=3 and 3 does not divide q. The resulting gap between the large general upper bound and the essentially p,q-independent lower bound motivates the unresolved asymptotic question.