Heights and quantitative arithmetic on stacky curves

Abstract

In this paper we investigate a family of algebraic stacks, the so-called stacky curves, in the context of the general theory of heights on algebraic stacks due to Ellenberg, Satriano, and Zureick-Brown. We first give an elementary construction of a height which is seen to be dual to theirs. Next we count rational points having bounded E-S-ZB height on a particular stacky curve, answering a question of Ellenberg, Satriano, and Zureick-Brown. We then show that when the Euler characteristic of stacky curves is nonpositive, that the E-S-ZB height coming from the anti-canonical divisor class fails to have the Northcott property. Next we prove a generalized version of a conjecture of Vojta, applied to stacky curves with negative Euler characteristic and coarse space P 1 , is equivalent to the abc-conjecture. Finally, we prove that in the negative characteristic case the purely "stacky" part of the E-S-ZB height exhibits the Northcott property.

Heights on stacks and a generalized Batyrev–Manin–Malle conjecture

“…As Nasserden and Xiao explain in [54,Theorem 1.4], the assertion that Conjecture 4.24 holds for all stacky curves is equivalent to the abc conjecture, with a key ingredient being a result of Granville [34]; indeed, Granville's result shows immediately that the two examples above satisfy Conjecture 4.24 conditional on abc. What is the relation between Vojta's 'more general abc conjecture' from [72] applied to a divisor D on a scheme X, and Conjecture 4.24 for a stack obtained by rooting a scheme X at D?6 One may hope that individual cases of Conjecture 4.24, like those described above, might not be as far out of reach as abc and its generalizations.…”

“…Starr and Xu [68,§1.4 of arXiv v1] have another definition whose relation to the one used in the present work is roughly that between the minimal slope in the Harder-Narasimhan filtration of a vector bundle and the slope of that vector bundle. And in very recent work, Nasserden and Xiao [54] offer an alternative definition for stacky curves, and Ratko Darda [20,Theorem 1.5.7.1] has proposed a definition for weighted projective stacks.…”