We provide a construction of free factorization algebras in algebraic geometry and link factorization homology of a scheme with coefficients in a free factorization algebra to the homology of its (unordered) configuration spaces. As an application, this construction allows for a purely algebro-geometric proof of homological stability of configuration spaces.

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Employing a geometric setting inspired by the proof of the Fundamental Lemma, we study some counting problems related to the average size of 2-Selmer groups and hence obtain an estimate for it.