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Polynomial partitioning techniques have recently led to improved geometric data structures for a variety of fundamental problems related to semialgebraic range searching and intersection searching in 3D and higher dimensions (e.g., see [Agarwal, Aronov, Ezra, and Zahl, SoCG 2019; Ezra and Sharir, SoCG 2021; Agarwal, Aronov, Ezra, Katz, and Sharir, SoCG 2022]). They have also led to improved algorithms for offline versions of semialgebraic range searching in 2D, via lens-cutting [Sharir and Zahl (2017)]. In this paper, we show that these techniques can yield new data structures for a number of other 2D problems even for online queries: 1) Semialgebraic range stabbing. We present a data structure for n semialgebraic ranges in 2D of constant description complexity with O(n^{3/2+ε}) preprocessing time and space, so that we can count the number of ranges containing a query point in O(n^{1/4+ε}) time, for an arbitrarily small constant ε > 0. (The query time bound is likely close to tight for this space bound.) 2) Ray shooting amid algebraic arcs. We present a data structure for n algebraic arcs in 2D of constant description complexity with O(n^{3/2+ε}) preprocessing time and space, so that we can find the first arc hit by a query (straight-line) ray in O(n^{1/4+ε}) time. (The query bound is again likely close to tight for this space bound, and they improve a result by Ezra and Sharir with near n^{3/2} space and near √n query time.) 3) Intersection counting amid algebraic arcs. We present a data structure for n algebraic arcs in 2D of constant description complexity with O(n^{3/2+ε}) preprocessing time and space, so that we can count the number of intersection points with a query algebraic arc of constant description complexity in O(n^{1/2+ε}) time. In particular, this implies an O(n^{3/2+ε})-time algorithm for counting intersections between two sets of n algebraic arcs in 2D. (This generalizes a classical O(n^{3/2+ε})-time algorithm for circular arcs by Agarwal and Sharir from SoCG 1991.)