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Abstract In this paper, we prove a uniform version of Poonen’s “Mordell-Lang Plus Bogomolov” theorem [ 12], based on Vojta’s method. Our main contribution is to generalize Rèmond’s work on the large points in order to allow an extra $$\epsilon $$-neighborhood in the canonical height topology. The part on small points follows from [ 8]. 

We introduce a variant of stable logarithmic maps, which we call punctured logarith- mic maps. They allow an extension of logarithmic Gromov–Witten theory in which marked points have a negative order of tangency with boundary divisors. As a main application we develop a gluing formalism which reconstructs stable logarithmic maps and their virtual cycles without expansions of the target, with trop- ical geometry providing the underlying combinatorics. Punctured Gromov–Witten invariants also play a pivotal role in the intrinsic con- struction of mirror partners by the last two authors, conjecturally relating to symplec- tic cohomology, and in the logarithmic gauged linear sigma model in work of Qile Chen, Felix Janda and Yongbin Ruan. 

We give a formula for the Chow rings of weighted blow-ups. Along the way, we also compute the Chow rings of weighted projective stack bundles, a formula for the Gysin homomorphism of a weighted blow-up, and a generalization of the splitting principle. In addition, in the Appendix we compute the Chern class of a weighted blow-up. 

We first introduce and study the notion of multi-weighted blow-ups, which is later used to systematically construct an explicit yet efficient algorithm for functorial logarithmic resolution in characteristic zero, in the sense of Hironaka. Specifically, for a singular, reduced closed subscheme $$X$$ of a smooth scheme $$Y$$ over a field of characteristic zero, we resolve the singularities of $$X$$ by taking proper transforms $$X_i \subset Y_i$$ along a sequence of multi-weighted blow-ups $$Y_N \to Y_{N-1} \to \dotsb \to Y_0 = Y$$ which satisfies the following properties: (i) the $$Y_i$$ are smooth Artin stacks with simple normal crossing exceptional loci; (ii) at each step we always blow up the worst singular locus of $$X_i$$, and witness on $$X_{i+1}$$ an immediate improvement in singularities; (iii) and finally, the singular locus of $$X$$ is transformed into a simple normal crossing divisor on $$X_N$$. Comment: Final published version 

In this paper, we extend a uniformity result of Dimitrov et al. [Uniformity in Mordell-Lang for curves, Ann. of Math. (2) 194(1) (2021) 237–298] to dimension two and use it to get a uniform bound on the cardinality of the set of all quadratic points for non-hyperelliptic non-bielliptic curves which only depend on the Mordell–Weil rank, the genus of the curve and the degree of the number field.