Existence of infinitely generated symbolic Rees rings of space monomial prime ideals in positive characteristic

Determine whether there exists a space monomial prime ideal p ⊂ K[x,y,z], where K is a field of positive characteristic and p is the defining ideal of the monomial curve (T^a, T^b, T^c) with pairwise coprime positive integers a, b, c, such that its symbolic Rees ring R_s(p) is not finitely generated over K. Equivalently, construct an explicit example of such a space monomial prime in positive characteristic or prove that all symbolic Rees rings of space monomial primes are finitely generated in positive characteristic.

Background

The paper studies finite generation of Cox rings of blow-ups of toric surfaces associated to triangles, relating them to extended symbolic Rees rings of certain ideals. A classical setting is when the ideal is the defining ideal of a space monomial curve (T^a, T^b, T^c) with pairwise coprime a, b, c; in this case the ideal is a space monomial prime and its symbolic Rees ring coincides with the Cox ring of a blow-up of the weighted projective plane P(a,b,c).

In characteristic zero, several criteria for finite generation are known. In positive characteristic, Cutkosky showed that the presence of a negative curve implies Noetherianity, but the existence of an example of a space monomial prime whose symbolic Rees ring is infinitely generated remains unsettled. Although this paper constructs infinitely generated examples in positive characteristic for certain non-prime ideals (and notes other non-monomial-prime constructions over finite fields), the specific case of space monomial prime ideals in positive characteristic remains open.