Lang–Vojta conjecture over function fields for surfaces dominating $${{\mathbb {G}}}_m^2$$

Abstract

We prove the nonsplit case of the Lang–Vojta conjecture over function fields for surfaces of log general type that are ramified covers of $${{\mathbb {G}}}_m^2$$ G m 2 . This extends the results of Corvaja and Zannier (J Differ Geom 93(3):355–377, 2013), where the conjecture was proved in the split case, and the results of Corvaja and Zannier (J Algebr Geom 17(2):295–333, 2008), Turchet (Trans Amer Math Soc 369(12):8537–8558, 2017) that were obtained in the case of the complement of a degree four and three component divisor in $${{\mathbb {P}}}^2$$ P 2 . We follow the strategy developed by Corvaja and Zannier and make explicit all the constants involved.

A truncated second main theorem for algebraic tori with moving targets and applications

“…The proof of Theorem 1.2 follows the ideas in [13] and [3] with extension to the moving situation. We note that Capuano and Turchet in [1] generalized the work of [3] to nonsplit function fields (that is, the moving case in our terminology) for surfaces. Finally, the proof of Theorem 1.3 is an adaption and generalization of Theorem 3 in [3] to the complex situation.…”

“…(a) if 𝑓 ∶ ℂ → 𝑋 be an algebraically nondegenerate analytic map, then 𝑁 (1) 𝑓 (𝐷, 𝑟) ⩾ exc 𝑇 𝐾+𝐷,𝑓 (𝑟) − o(𝑇 𝐴,𝑓 (𝑟));…”

“…(1.1) (b) for any 𝜖 > 0, there is a proper Zariski-closed subset 𝑍 of 𝑋, depending only on 𝑋, 𝐷, 𝐴, and 𝜖 such that for any analytic map 𝑓 ∶ ℂ → 𝑋 whose image is not contain in 𝑍, the following 𝑁 (1) 𝑓 (𝐷, 𝑟) ⩾ exc 𝑇 𝐾+𝐷,𝑓 (𝑟) − 𝜖𝑇 𝐴,𝑓 (𝑟) (1.2)…”

“…For example, the conjecture holds for dim 𝑋 = 1. The Cartan's second main theorem, where 𝑋 = ℙ 𝑛 and 𝐷 = 𝐻 1 + ⋯ + 𝐻 𝑞 , where the 𝐻 𝑖 are hyperplanes in general position, suggests that (1.1) holds with 𝑁 (1) 𝑓 (𝐷, 𝑟) replaced by ∑ 𝑞 𝑖=1 𝑁 (𝑛) 𝑓 (𝐻 𝑖 , 𝑟) under a weaker assumption that the map 𝑓 is linearly nondegenerate. When 𝑋 is a semiabelian variety, Noguchi, Winkleman and Yamanoi in [14] showed that the inequality (1.1) holds with 𝑁 (1) 𝑓 (𝐷, 𝑟) replaced by 𝑁 (𝑘 0 ) 𝑓 (𝐷, 𝑟) for some positive integer 𝑘 0 , and (1.2) holds if the map is algebraically nondegenerate.…”

“…Let 𝐺 be a nonconstant homogeneous polynomial in 𝐾 𝐮 [𝑥 0 , … , 𝑥 𝑛 ] with no repeated nonmonomial factors in 𝐾 𝐮 [𝑥 0 , … , 𝑥 𝑛 ]. If 𝑢 0 , … , 𝑢 𝑛 are multiplicatively independent modulo 𝐾 𝐮 , then for any 𝜖 > 0 𝑁 𝐺(𝐮) (0, 𝑟) − 𝑁 (1) 𝐺(𝐮) (0, 𝑟) ⩽ exc 𝜖𝑇 𝐮 (𝑟).…”